Earth-ionosphere waveguide power transfer

ABSTRACT

Systems and methods for detecting low-loss eigenmodes of a spherical waveguide bounded by the Earth&#39;s surface and its ionosphere are disclosed. One or more eigenmodes of the Earth-ionosphere waveguide may be computed based on a mathematical model incorporating electrical properties of the terrestrial surface and plasma physics of the ionospheric layer. A transmitter apparatus may be used transmit electrical power into the Earth-ionosphere waveguide in the form of an electromagnetic wave, which may, in turn, be detected by a receiver apparatus remote from the transmitter apparatus. A coupling strength between the transmitted electromagnetic wave and the one or more eigenmodes may be determined by measuring power received by the receiver apparatus in the detected electromagnetic wave. By iteratively adjusting parameters of the transmitter apparatus, determining the coupling strength, and refining a quantitative description of the eigenmodes, the eigenmodes may be harnessed for wireless power transmission throughout the Earth-ionosphere waveguide.

BACKGROUND

Waveguides can be used for electromagnetic power transmission. Inparticular, the geometry and physical-electromagnetic properties of awaveguide may determine modes of electromagnetic propagationcorresponding to persistent or semi-persistent solutions to Maxwell'sEquations, subject to boundary conditions of the waveguide. Suchmathematical solutions are referred to formally as “eigenmodes,” andrepresent resonant modes of the waveguide that can give rise to standingwaves within the waveguide, a volume customarily referred to as thewaveguide “cavity.” Introducing or injecting electromagnetic energy intoa waveguide cavity in such a way that “excites” or “couples to” aneigenmode causes power to be transmitted efficiently within thewaveguide, without a physical transmission medium, such as wires.

SUMMARY

In one aspect, an example embodiment presented herein provides a methodcomprising: transmitting, by a transmitter apparatus, electrical powerinto a spherical waveguide bounded by a terrestrial surface and anionospheric layer, wherein the electrical power is transmitted in anelectromagnetic wave; computing one or more eigenmodes of the sphericalwaveguide based on a mathematical model of the spherical waveguide thatincorporates electrical properties of the terrestrial surface and plasmaphysics of the ionospheric layer; detecting the transmittedelectromagnetic wave by a receiver apparatus remote from the transmitterapparatus; and determining a strength of coupling between thetransmitted electromagnetic wave and the one or more eigenmodes bymeasuring an amount of power received by the receiver apparatus in thedetected electromagnetic wave.

In another aspect, an example embodiment presented herein provides asystem comprising: a transmitter apparatus; a receiver apparatus remotefrom the transmitter apparatus; and a computer apparatus having one ormore processors and memory storing instructions that, when executed bythe one or more processors, cause the system to carry out operationsincluding: causing the transmitter apparatus to transmit electricalpower into a spherical waveguide bounded by the terrestrial surface ofthe Earth and the ionospheric layer of the Earth, wherein the electricalpower is transmitted in an electromagnetic wave; computing one or moreeigenmodes of the spherical waveguide based on a mathematical model ofthe spherical waveguide that incorporates electrical properties of theterrestrial surface and plasma physics of the ionospheric layer; causingthe receiver apparatus to detect the transmitted electromagnetic wave;and determining a strength of coupling between the transmittedelectromagnetic wave and the one or more eigenmodes by measuring anamount of power received by the receiver apparatus in the detectedelectromagnetic wave.

In yet another aspect, an example embodiment presented herein provides asystem comprising: a transmitter apparatus; a receiver apparatus remotefrom the transmitter apparatus; and a computer apparatus having one ormore processors and memory storing instructions that, when executed bythe one or more processors, cause the system to carry out operationsincluding: causing the transmitter apparatus to transmit electricalpower into a spherical waveguide bounded by the terrestrial surface ofthe Earth and the ionospheric layer of the Earth, wherein the electricalpower is transmitted in an electromagnetic wave; computing one or moreeigenmodes of the spherical waveguide based on a mathematical model ofthe spherical waveguide that incorporates electrical properties of theterrestrial surface and plasma physics of the ionospheric layer; causingthe receiver apparatus to detect the transmitted electromagnetic wave;based on the determined one or more eigenmodes and an amount of powerreceived by the receiver apparatus in the detected electromagnetic wave,adjusting at least one of a frequency, amplitude, or phase of theelectrical power transmitted by the transmitter apparatus so as to causea predicted change in the amount of power received by the receiverapparatus in the detected electromagnetic wave; and determining whetheror not a measured change in power received at the receiver apparatus iswithin a threshold of the predicted change in received power.

These as well as other aspects, advantages, and alternatives will becomeapparent to those of ordinary skill in the art by reading the followingdetailed description with reference where appropriate to theaccompanying drawings. Further, it should be understood that thedescription provided in this summary section and elsewhere in thisdocument is intended to illustrate the claimed subject matter by way ofexample and not by way of limitation.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a computing system, in accordance withexample embodiments.

FIG. 2 is a simplified diagram or a computer network, in accordance withexample embodiments.

FIG. 3A is a conceptual illustration of the ionosphere, showing day andnight sides of the Earth.

FIG. 3B is an enlarged view of a portion of the conceptual illustrationof the ionosphere shown in FIG. 3A.

FIG. 4A is a conceptual illustration of low frequency radio propagationin the presence of the D layer of the ionosphere.

FIG. 4B is a conceptual illustration of low frequency radio propagationin the presence of a diminished D layer of the ionosphere.

FIG. 4C is a conceptual illustration of low frequency radio propagationin the absence of the D layer of the ionosphere.

FIG. 5 is a conceptual illustration of a simulation, in accordance withexample embodiments.

FIG. 6 is a conceptual illustration of a simulation, in accordance withexample embodiments.

FIG. 7 is a conceptual illustration of a simulation, in accordance withexample embodiments.

FIG. 8 is a conceptual illustration of a simulation, in accordance withexample embodiments.

FIG. 9 is a conceptual illustration of a simulation, in accordance withexample embodiments.

FIGS. 10A, 10B, and 10C each depict a conceptual illustration of adipole field, in accordance with example embodiments.

FIG. 11 is a simplified illustration of a transmitter antenna array, inaccordance with example embodiments.

FIG. 12 is a simplified illustration of a receiver antenna array, inaccordance with example embodiments.

FIG. 13 is a simplified illustration of a transmitter antenna array anda distribution of remotely located receiver devices, in accordance withexample embodiments.

FIG. 14A is flowchart of one example method of determining and couplingto an eigenmode, in accordance with example embodiments.

FIG. 14B is flowchart of another example method of determining andcoupling to an eigenmode, in accordance with example embodiments.

FIG. 15 is a simplified illustration of a helical antenna, in accordancewith example embodiments.

FIG. 16A is an illustration of an antenna pattern, in accordance withexample embodiments.

FIG. 16B is an illustration of a different antenna pattern, inaccordance with example embodiments.

FIG. 17 is a simplified illustration of a helical antenna havingvariable dimensional parameters, in accordance with example embodiments.

FIG. 18A is an example transmitter helical coupler, in accordance withexample embodiments.

FIG. 18B is another example transmitter helical coupler, in accordancewith example embodiments.

FIG. 18C is still another example transmitter helical coupler, inaccordance with example embodiments.

FIG. 18D is yet another an example transmitter helical coupler, inaccordance with example embodiments.

FIG. 19 is a simplified illustration of a helical supergain antenna, inaccordance with example embodiments.

FIG. 20 is block diagram of a waveguide coupler, in accordance withexample embodiments.

FIG. 21 is flowchart of an example method of coupling to an eigenmodeusing an antenna array, in accordance with example embodiments.

FIG. 22 is block diagram of a wireless power distribution system, inaccordance with example embodiments.

FIG. 23A illustrates a transmitter helical coupler, in accordance withexample embodiments.

FIG. 23B illustrates another transmitter helical coupler, in accordancewith example embodiments.

FIG. 24 is a conceptual illustration of standing waves in anEarth-ionosphere waveguide, in accordance with example embodiments.

FIG. 25 is flowchart of an example method of coupling to an eigenmodeusing an array of antenna arrays, in accordance with exampleembodiments.

FIG. 26 is flowchart of an example method of detecting the presence of aload, in accordance with example embodiments.

FIG. 27 is flowchart of an example method adjusting the power level at aspecific location, in accordance with example embodiments.

DETAILED DESCRIPTION

Exemplary methods and systems are described herein. It should beunderstood that the word “exemplary” is used herein to mean “serving asan example, instance, or illustration.” Any implementation or featuredescribed herein as “exemplary” or “illustrative” is not necessarily tobe construed as preferred or advantageous over other implementations orfeatures. In the figures, similar symbols typically identify similarcomponents, unless context dictates otherwise. The exampleimplementations described herein are not meant to be limiting. It willbe readily understood that the aspects of the present disclosure, asgenerally described herein, and illustrated in the figures, can bearranged, substituted, combined, separated, and designed in a widevariety of different configurations, all of which are contemplatedherein.

I. Overview

Wireless transmission of electromagnetic power is generally limited bythe effects of propagating electromagnetic waves in free space, whichradiate power away at distances beyond roughly the wavelength of thetransmission source. This makes broadly distributed wireless powertransmission impractical without some means for coupling to theelectromagnetic wave energy within less than about a wavelength from thesource. Using waveguides can allow such coupling, while effectivelyextending the range of transmission by way of interaction of the wavewith the waveguide, at least across the physical extent of the waveguidecavity. The physical extent of the waveguide thus represents dimensionsacross which power may be transmitted wirelessly. In this sense, awaveguide replaces wires for transmission, but may still be similarlylimited in terms of physical distances of transmission and distributionof power by virtue of the physical extent of the waveguide.

In example embodiments, portions of the Earth itself, including theionosphere, can be treated as a waveguide, and as such, wirelesstransmission of electromagnetic power may be carried out on a physicalscale comparable to the entire Earth. More specifically, by accountingfor electrical physical properties of the terrestrial surface of theEarth and for plasma physics of the Earth's ionosphere, the spacebetween these two layers (i.e., the Earth's surface and the ionosphere)can be modeled as a spherical waveguide or waveguide cavity, and itselectromagnetic properties analyzed. In particular, computations,including simulations, may be used to determine eigenmodes of this“whole Earth” or Earth-ionosphere waveguide. It will be appreciated thatneither the Earth nor the ionosphere is perfectly spherical. As such,the Earth-ionosphere waveguide cavity also is not perfectly spherical.Geometrically, however, the deviations from perfect sphericity aregenerally negligible compared with the dimensions of the Earth and theionosphere. Accordingly, references to a spherical waveguide orwaveguides herein should be understood to encompass both perfectsphericity and deviations from perfect sphericity on the order of thosecomparable to the Earth and its ionosphere.

In accordance with example embodiments, the electrical physicalproperties of the Earth can be approximated and/or modeled as largelyconsisting of defined regions of wet and dry land, sea water, and/orfresh water, each region having specifiable permittivity andconductivity. Still other regions having different physicalcharacteristics and/or material constituent properties and electricalproperties may be incorporated into a model of the Earth's terrestrialsurface as well. The model may approximate the surface as spherical (ornearly so), and may include topographical components or features aswell.

Also in accordance with example embodiments, one or more existing modelsof the Earth ionosphere may be utilized to determine physical andbehavioral properties of the ionosphere that manifest as an effectiveouter or upper boundary of the Earth-ionosphere waveguide cavity, atleast for one or more transmission frequency ranges. Such existingmodels may represent the ionosphere according to appropriate plasmaphysics, taking into account factors such as electron collisions withneutral particles (atoms and molecules), the gyrotropic properties dueto the Earth's magnetic field, and the number density of electrons andpossibly other species as a function of location and time. Time andlocation dependencies may, at least in part, derive from diurnalvariations of solar and other excitation sources that partially ionizeand create the plasma out of the atmospheric gas mixture, given that gasmixture and density as a function of position. Example properties mayinclude effective electrical conductivity and/or permittivity as afunction of frequency, altitude (height above the surface), Earth'smagnetic field direction and magnitude, geographic location, and time ofday.

Taking the Earth-ionosphere waveguide as represented by a cavity boundedbetween the terrestrial surface model and the ionosphere model, one ormore eigenmodes may be determined by solving Maxwell's Equations subjectto the boundary conditions of the cavity. In an example embodiment,solutions to Maxwell's Equations applied to the Earth-ionospherewaveguide may be determined using one or more simulation tools. In someembodiments, the simulation tools may parameterize a physical system toderive numerical solutions of analytical equations representing thephysical system. More particularly, simulation tools may be applied tothe Earth-ionosphere waveguide.

In an example embodiment, simulation results may yield predictedeigenmodes that propagate with spatial periodicity (e.g., wavelengths)and electromagnetic form within the Earth-ionosphere waveguide cavity soas to give rise to low-loss standing or traveling waves. The predictedexistence of these eigenmodes, as well as the techniques for making thepredictions, can provide a basis for analytically-guided empiricalmethods and systems for exciting one or more of the eigenmodes. Byexciting the one or more eigenmodes, electromagnetic power may bewirelessly transmitted within at least a portion of the Earth-ionospherewaveguide cavity. These eigenmodes are low-loss in the sense that only arelatively small portion of the power that couples to the eigenmodes issubject to electromagnetic loss processes within the Earth-ionospherewaveguide cavity, although there can be some geometric dilution of poweras a function of distance from a source of excitation. Exampleembodiments disclosed herein of analytically-guided empirical systemsand methods for exciting one or more of eigenmodes may therefore be usedto wirelessly transmit electromagnetic power between any two or morepoints along or within the waveguide cavity. Example embodiments ofexcitation systems and methods also may serve to confirm the existenceof one or more eigenmodes, and to discern and validate physical andquantitative properties of the eigenmodes.

The predicted eigenmodes have certain characteristics similar to thoseof previous solutions of Maxwell's Equations that have been derived byconsidering less comprehensive representations of the Earth environment,or by considering only certain aspects of the Earth environment. But therestrictions of previous analyses and modeling in terms of detail and/orphysical regimes considered have left the low-loss eigenmodesundiscovered or simply beyond the realm of exploration. In this sense,the advantages of detailed modeling of the Earth-ionosphere waveguidecavity in accordance with example embodiments are therefore clearlydemonstrated.

In addition to example embodiments for systems and methods for detectingand confirming low-loss eigenmodes inferred or predicted from detailedmodeling techniques, example embodiments are also described herein forother aspects and applications of wireless power transmission in theEarth-ionosphere waveguide. In particular, other example embodiments aredescribed for: configurations and use of antenna arrays to efficientlyexcite and couple to one or more eigenmodes of the Earth-ionospherewaveguide at one location, and to efficiently receive power from one ormore eigenmodes at another location; configurations and use of arrays ofantenna arrays to excite one or more eigenmodes in a globally-phasedmanner to generate standing or traveling waves and make the power thewaves carry available on a global basis; configurations and use ofarrays of antenna arrays to excite one or more eigenmodes in aglobally-phased manner to detect and determine locations of loads thatrepresent sinks/taps of power; and configurations and use of arrays ofantenna arrays to excite one or more eigenmodes in a globally-phasedmanner to direct or “steer” nulls or other controlled levels of energydensity to one or more specific locations. It will be appreciated thatthese specific aspects do not represent an exhaustive or limiting listof aspects of wireless power transmission in the Earth-ionospherewaveguide disclosed herein, and that other aspects, whether expresslyidentified or not, may be represented—explicitly or implicitly—as well.

Example Computing System and Network

Example methods, devices, systems and apparatuses described herein maybe implemented, at least in part, with or in the context of one or morecomputing systems and one or more computer networks. Example computingsystems and networks may be configured and/or constructed using avariety of hardware, software, and firmware components. FIGS. 1 and 2illustrate example computing systems and networks.

FIG. 1 is a simplified block diagram showing some of the components ofan example computing device 100. As shown in FIG. 1, computing device100 may include a communication interface 102, a user interface 104, aprocessor 106, and data storage 108, all of which may be communicativelylinked together by a system bus, network, or other connection mechanism110.

Communication interface 102 functions to allow computing device 100 tocommunicate, using analog or digital modulation, with other devices,access networks, and/or transport networks. Thus, communicationinterface 102 may facilitate circuit-switched and/or packet-switchedcommunication, such as POTS communication and/or IP or other packetizedcommunication. For instance, communication interface 102 may include achipset and antenna arranged for wireless communication with a radioaccess network or an access point. Also, communication interface 102 maytake the form of a wireline interface, such as an Ethernet, Token Ring,or USB port. Communication interface 102 may also take the form of awireless interface, such as a Wifi, BLUETOOTH®, global positioningsatellite (GPS) system, or wide-area wireless interface (e.g., WiMAX orLTE). A GPS system could be used for example to provide precise timingand/or geolocation information for various applications or functionsdescribed herein that may use or need such information. However, otherforms of physical layer interfaces and other types of standard orproprietary communication protocols may be used over communicationinterface 102. Furthermore, communication interface 502 may comprisemultiple physical communication interfaces (e.g., a Wifi interface, aBLUETOOTH® interface, and a wide-area wireless interface).

User interface 104 may function to allow computing device 100 tointeract with a human or non-human user, such as to receive input from auser and to provide output to the user. Thus, user interface 104 mayinclude input components such as a keypad, keyboard, touch-sensitive orpresence-sensitive panel, computer mouse, trackball, joystick,microphone, still camera and/or video camera. User interface 104 mayalso include one or more output components such as a display screen(which, for example, may be combined with a touch-sensitive panel), CRT,LCD, LED, a display using DLP technology, printer, light bulb, and/orother similar devices, now known or later developed. User interface 104may also be configured to generate audible output(s), via a speaker,speaker jack, audio output port, audio output device, earphones, and/orother similar devices, now known or later developed. In someembodiments, user interface 504 may include software, circuitry, oranother form of logic that can transmit data to and/or receive data fromexternal user input/output devices. Additionally or alternatively,computing device 100 may support remote access from another device, viacommunication interface 102 or via another physical interface (notshown).

Processor 106 may comprise one or more general purpose processors (e.g.,microprocessors) and/or one or more special purpose processors (e.g.,DSPs, GPUs, FPUs, network processors, or ASICs). Data storage 108 mayinclude one or more volatile and/or non-volatile storage components,such as magnetic, optical, flash, or organic storage, and may beintegrated in whole or in part with processor 106. Data storage 108 mayinclude removable and/or non-removable components.

In general, processor 106 may be capable of executing programinstructions 118 (e.g., compiled or non-compiled program logic and/ormachine code) stored in data storage 108 to carry out the variousfunctions, operations, and or method steps described herein. Therefore,data storage 108 may include a non-transitory computer-readable medium,having stored thereon program instructions that, upon execution bycomputing device 100, cause computing device 100 to carry out any of themethods, processes, or functions disclosed in this specification and/orthe accompanying drawings. Non-limiting examples of non-transitoryinstructions include software, firmware and hardware instructions. Theexecution of program instructions 118 by processor 106 may result inprocessor 106 using data 112.

By way of example, program instructions 118 may include an operatingsystem 122 (e.g., an operating system kernel, device driver(s), and/orother modules) and one or more application programs 120 (e.g., addressbook, email, web browsing, social networking, and/or gamingapplications) installed on computing device 100. Similarly, data 112 mayinclude operating system data 116 and application data 114. Operatingsystem data 116 may be accessible primarily to operating system 122, andapplication data 114 may be accessible primarily to one or more ofapplication programs 120. Application data 114 may be arranged in a filesystem that is visible to or hidden from a user of computing device 100.

Application programs 120 may communicate with operating system 122through one or more application programming interfaces (APIs). TheseAPIs may facilitate, for instance, application programs 120 readingand/or writing application data 114, transmitting or receivinginformation via communication interface 102, receiving or displayinginformation on user interface 104, and so on.

In some vernaculars, application programs 120 may be referred to as“apps” for short. Additionally, application programs 120 may bedownloadable to computing device 100 through one or more onlineapplication stores or application markets. However, application programscan also be installed on computing device 100 in other ways, such as viaa web browser or through a physical interface (e.g., a USB port) oncomputing device 100.

FIG. 2 is a simplified block diagram of a communication system 200, inwhich various embodiments described herein can be employed, or which mayserve one or another communication function described herein.Communication system 200 includes computing devices 202, 204, 206 and208, each of which could be a computing device such as or similar to thecomputing device 100 described above. Each of these computing devicesmay be able to communicate with other devices (including with eachother) via a network 210 through the use of wireline connections(designated by solid lines) and/or wireless connections (designated bydashed lines).

Network 210 may be, for example, the Internet, or some other form ofpublic or private Internet Protocol (IP) network. Thus, computingdevices 202, 204, 206 and 208 may communicate using packet-switchingtechnologies. Nonetheless, network 210 may also incorporate at leastsome circuit-switching technologies, and computing devices 202, 204, 206and 208 may communicate via circuit switching alternatively or inaddition to packet switching.

II. Example Systems and Methods for Predicting and Detecting Eigenmodesof the Earth-Ionosphere Waveguide

Achieving efficient transmission of power in a waveguide depends, atleast in part, on coupling electromagnetic energy to one or moreeigenmodes of the waveguide. In a typical case in which a waveguide maybe constructed according to specifications, achieving such coupling canbe reasonably assured by design. Additionally, when design andconstruction of the waveguide are possible, electrical properties of thewaveguide boundaries may be controlled to help minimize transmissionlosses. For the endeavor of treating the volume bounded by Earth'ssurface and its ionosphere as a waveguide, the form and electricalproperties of the waveguide occur naturally and are not subject todesign. Rather, techniques and procedures are needed for realizingwireless power transmission using the Earth-ionosphere waveguide andmaking it practical. Accordingly, example embodiments are disclosedherein for systems and methods that implement an approach to achievingwireless transmission of electromagnetic power in the Earth-ionospherewaveguide cavity that occupies the volume bounded by Earth's surface andits ionosphere that involves an analytic aspect as well as an empiricalaspect. For purposes of the discussion herein, these aspects will begenerally referred to as “components” of the approach. It should beunderstood, however, that the description of the approach in terms of“components” is a convenience for organizing the discussion, and notintended to be limiting with respect to disclosed example embodiments.Continuing then, an analytic component provides for both predicting whatthe eigenmodes look like and predicting a degree of coupling betweentransmitted power of a given excitation source and the eigenmodes.Further, an empirical component is used for exciting and coupling to oneor more of the predicted eigenmodes, detecting and receiving power fromone or more of the excited eigenmodes, and validating and refininganalytic descriptions of the eigenmodes based on controlled and/orobserved characteristics of transmitted and received power.

The existence of low-loss eigenmodes of the Earth-ionosphere waveguidemay be based, in part, on electrical properties of the Earth's surfaceand plasma physics of the ionosphere, as well as the known existenceand/or study of other propagation modes having characteristics thatsuggest or indicate that electromagnetic transmission properties of theEarth would support low-loss eigenmodes as well. A brief summary reviewof the Earth's surface, plasma physics of the ionosphere, and some ofthese other propagation modes helps provide a context for the techniquesfor making analytical predictions of low-loss eigenmodes.

Electrical Properties of the Earth's Surface.

Electrical properties of the Earth's surface generally refer to groundconductivities, and can be presented in the form of conductivity maps. Astandard source of ground conductivities is the World Atlas of GroundConductivities (Recommendation ITU-R P.832-4). These maps, shown inunits of conductivity in the report are mS/m=10³ μS/m, or milli-siemensper meter, apply to radio frequencies in a VLF (very low frequencyrange), up to 30 kHz, and a MF (medium frequency) range standardized to1 MHz.

In the context of the present disclosure, the conductivity maps can beused to make gross estimates that treat the Earth's surface ashomogeneous, having average conductivity properties. For example, aqualitative examination of the conductivity maps supports a reasonableassumption of that Earth surface is ⅔ sea water and ⅓ land, and that ⅓of land is good conductor, ⅔ of land is bad conductor. From this,estimated “uniform” ground conductivities of 0.01-4 S/m may be inferred.This broad range provides reasonable bounds on low-end and high-endconductivities for the uniform estimate. For some of the modelingdescribed below, a single, homogeneous conductivity value may sufficefor investigative stages of the techniques described.

The conductivity maps may be used in more detailed applications of themodels to determine spatial distributions of ground conductivities. Forexample, the models could incorporate non-uniform surface distributions.Doing so may allow the effect of land-ocean boundaries to be included inthe eigenmode solutions. As described later, accounting for spatialdiscontinuities in ground conductivities, both in modeling and empiricalmeasurements, can help guide operational aspects of power transmission.

Electromagnetic and Dynamical Properties of the Ionosphere.

The ionosphere is a layer of the Earth's atmosphere between about75-1000 km above the surface that is ionized by solar and cosmicradiation. It consists predominantly of three layers, conventionallyreferred to as the D, E, and F layers. The layers are characterizedmainly by their degree of ionization, which, in turn, depends on theirexposure to the sources of ionizing radiation. The F layer, the highestand most exposed layer, has the highest degree of ionization. It ispresent during both daytime and nighttime. The E layer, below the Flayer, receives less ionizing radiation, and is therefore less ionized.During the nighttime, it becomes weakened (its ionization decreases).The D layer, the lowest of the three layers, is the least ionized. Itlargely disappears during the nighttime.

FIG. 3A shows a conceptual illustration of the ionosphere, depicting adaytime and nighttime side of the Earth. The daytime side shows the Dand E layers, and the nighttime side shows just the E layer. F layer isomitted for clarity. FIG. 3B shows an enlarged view a portion of theillustration of FIG. 3A. In this view, the D layer 306 and E layer 304can be clearly distinguished. The F layer has again been omitted forclarity.

The ionosphere influences radio frequency (RF) propagation betweendistant terrestrial locations and between the Earth's surface andsatellites beyond the ionosphere. At very high radio frequencies (e.g.several MHz), radiation may propagate through the entire ionosphere withlittle or no attenuation. At low frequencies (LF), some or all energy ofincident radiation may be lost due to interactions in the ionizedplasma. Specifically, the E layer will generally reflect incident lowfrequency waves, thereby acting like a waveguide enabling propagationaround some portion of the globe. Depending on frequency, the D layerwill act to attenuate incident low frequency radiation, consuming someof its energy. Some portion of the incident radiation may make it to theE layer and reflect back for a return traversal of the D layer andfurther attenuation. Depending on the thickness of the D layer, whichvaries over the course of the day as described above, the reflected wavemay then propagate back to the Earth's surface. When the D layer isthickest (roughly midday), all of the incident radiation may be lost tothe D layer. When the D layer is thinnest or absent, some or most of theradiation may be reflected back.

FIGS. 4A, 4B, and 4C illustrate the effects of the D and E layers on LFradiation. An arrow in FIG. 4A represents LF radiation directed from theEarth's surface 402 to the ionosphere during daytime when the D layer406 is thickest (and extends to its lowest altitude). The LF radiationpenetrates the D layer and is attenuated before reaching the E layer404. FIG. 4B represents a time at which the D layer is thinner, thoughstill present. In this case, the LF radiation makes it to the E layerand is reflected, though it will suffer attenuation on both its upwardand downward (reflected) paths. FIG. 4C represent nighttime, when the Dlayer is absent. In the case, the LF radiation is reflected directly offthe E layer without any losses to the D layer.

The degree of ionization in the different ionosphere layers can also berepresented as different values of electrical conductivity. As such, theionosphere can act like a waveguide boundary or wall characterized byconductivity. Depending on the frequency of the radiation and thediurnally-dependent conductivity, the waveguide wall may presentdifferent degrees of reflectivity and absorption. This dynamic behavioris governed by the plasma physics of the ionosphere, which can bemathematically modeled using various techniques. One widely used modelis the International Reference Ionosphere (IRI) World Atlas of GroundConductivities (available fromhttps://omniweb.gsfc.nasa.gov/vitmo/iri2012_vitmo.html), which is basedon electron density, electron and ion temperature, and the ioniccomposition. Other models may be available as well.

In the context of the present disclosure, an ionosphere model, such asthe IRI model, may be included in simulations used to derive eigenmodesof the Earth-ionosphere waveguide, as described in detail below. As alsodiscussed below, accounting for the presence and behavior of theionosphere is advantageous to understanding how low-loss eigenmodes maymanifest.

Zenneck Surface Wave Mode.

Zenneck waves are solutions of Maxwell's Equations for wave propagationover a surface that is a boundary between a first region below thesurface and a second region above. The two regions are characterized bydifferent electrical properties, such as conductivity and permittivity.For an electrically lossy first region, an environment that is generallydescriptive of the terrestrial surface of the Earth, the solutions yieldelectromagnetic waves that are guided along the surface as theypropagate. By way of reference, a set of solutions derived by Barlow andCullen (Barlow, H. M., and Cullen, A. L., “Surface Waves,” Proceedingsof the IEE—Part III: Radio and Communication Engineering, Volume 100,Issue 68, November 1953) in cylindrical coordinates (r,z,ϕ), where r isa radial coordinate parallel to the surface, z is a vertical coordinate,and ϕ is an azimuthal angle about z, can be described by Hankelfunctions in the radial direction and by exponential decay in thepositive and negative vertical (±z) directions.

The solutions inside (below) the surface (z≤0) are given by Equations1a, 1b, and 1c:

$\begin{matrix}{H_{\varphi \; 1} = {A\; e^{i\; \omega \; t}e^{u_{1}z}{H_{1}^{(2)}\left( {{- i}\; \gamma \; r} \right)}}} & \left\lbrack {1a} \right\rbrack \\{E_{r\; 1} = {{- {A\left( \frac{u_{1}}{\sigma_{1} + {i\; \omega \; \kappa_{1}}} \right)}}e^{i\; \omega \; t}e^{u_{1}z}{H_{1}^{(2)}\left( {{- i}\; \gamma \; r} \right)}}} & \left\lbrack {1b} \right\rbrack \\{E_{z\; 1} = {{A\left( \frac{i\; \gamma}{\sigma_{1} + {i\; \omega \; \kappa_{1}}} \right)}e^{i\; \omega \; \kappa}e^{u_{1}z}{{H_{0}^{(2)}\left( {{- i}\; \gamma \; r} \right)}.}}} & \left\lbrack {1c} \right\rbrack\end{matrix}$

The solutions outside (above) the surface (z≥0) are given by Equations2a, 2b, and 2c:

$\begin{matrix}{H_{\varphi \; 2} = {A\; e^{i\; \omega \; t}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- i}\; \gamma \; r} \right)}}} & \left\lbrack {2a} \right\rbrack \\{E_{r\; 2} = {{A\left( \frac{u_{2}}{i\; \omega \; \kappa_{0}} \right)}e^{i\; \omega \; t}e^{{- u_{2}}z}{H_{1}^{(2)}\left( {{- i}\; \gamma \; r} \right)}}} & \left\lbrack {2b} \right\rbrack \\{E_{z\; 2} = {{A\left( \frac{\gamma}{\kappa_{0}\;} \right)}e^{i\; \omega \; t}e^{{- u_{2}}z}{{H_{0}^{(2)}\left( {{- i}\; \gamma \; r} \right)}.}}} & \left\lbrack {2c} \right\rbrack\end{matrix}$

In these equations, H_(ϕ1) and H_(ϕ2) are the transverse (azimuthal)components of the magnetic field H in region 1 (below the surface) andregion 2 (above the surface), respectively; E_(r1) and E_(r2) are theradial components of the electric field E in regions 1 and 2,respectively, and E_(z1) and E_(z2) are the vertical components of E inregions 1 and 2, respectively. The conductivity and permittivity inregion 1 are σ₁ and κ₁, respectively, and the permittivity of region 2is taken to that of free space, κ₀. The frequency of the wave is f=ω/2π,the vertical propagation constants are u₁ and u₂ in regions 1 and 2,respectively, and the radial propagation constant is y.

From the solutions above, Zenneck waves have a Poynting vector directedslightly downward (into the surface) but mostly in the outward radialdirection. The field has a transverse magnetic component, and isevanescent in the direction of propagation with a geometric factor of1/√{square root over (r)} due to being bound to the surface. At lowfrequencies (e.g., 10s of kHz) the 1/√{square root over (r)} geometricfactor dilutes the power density faster than the exponential decayattenuates it. The radial exponential decay may not be a limitingpractical factor in power transmission. In the direction normal to thesurface, the wave is subject to total internal reflection, and decaysexponentially in the vertical direction above the surface. Theexponential vertical component feeds ohmic or I²R losses so that, to theextent that the scale height of the decay is smaller than a height of anelectrically lossy boundary surface, I²R losses may not be significant.

Among the characteristics of Zenneck surface wave solutions that helpinform the techniques for predicting low-loss eigenmodes of theEarth-ionosphere waveguide are: (i) the indication that the terrestrialsurface of the Earth can support propagation of a trappedelectromagnetic surface wave, and (ii) the indication that forlow-frequency Zenneck waves—the ones most subject to surfacetrapping—the vertical scale height of the wave extends at least to thelower layers of the ionosphere, so that I²R losses do actually becomesignificant. The first characteristic indicates that the Earth's surfacecan be modeled as the inner boundary of a spherical waveguide cavity.Note that the more lossy the surface, the more constrained the wave isto the surface. This has implications when considering Zenneck wavespropagating on a spherical surface. The second characteristic suggeststhat among various modes to which a source transmitter may couple,Zenneck waves are not likely to primarily or substantially supportlow-loss power transmission owing to I²R losses in the ionosphere.

Schumann cavity resonance. Schumann resonance is an electromagneticphenomenon in which atmospheric lightning discharges excite resonantmodes of a waveguide cavity formed by the Earth's surface and itsionosphere. The existence of such resonant modes, first predictedmathematically by Schumann in 1952 and later detected in the early 1960s(see, for example, “Schumann resonances” athttps://en.wikipedia.org/wiki/Schumann_resonances), providesconfirmation that an Earth-ionosphere waveguide is more than atheoretical construct. In this sense, the existence of Schumannresonance also provides strong support for the existence of low-losseigenmodes. But, the physical characteristics of Schumann resonances andof lightning that excites them also indicate that they are not likely tobe modes that can be harnessed for wireless power transmission.Nevertheless, an understanding of Schumann resonances can be instructiveand inform the analytical and empirical techniques for wireless powertransmission disclosed herein.

The modes excited by lightning are predominantly the lowest resonantfrequency modes derived by Schumann. For an ideal spherical waveguidecavity, the resonant frequency f_(n) of the n^(th) mode is given by

${f_{n} = {\frac{c}{2\pi \; a}\sqrt{n\left( {n + 1} \right)}}},$

where a is the radius of the inner sphere (e.g., the Earth's surface inthe case of Schumann resonance) and c is the speed of light. In the caseof the actual Earth-ionosphere waveguide, the speed of light is lowerdue partly to losses in the ionosphere. As a consequence of this, amongother non-ideal aspects of the real conditions, the observed resonantfrequencies of Schumann resonances are lower than in the ideal case,with detections of the first few modes at extremely low frequencies(ELFs) of 7.83, 14.3, 20.8, 27.3, and 33.8 Hz.

For a resonant cavity in general, a quality factor Q may be defined as aratio of energy stored in the cavity to energy lost per cycle in thecavity walls. As such, Q measures loss characteristics of the cavity,where losses are typically ohmic. For a low-loss cavity, Q is high, andvice versa. It can be shown in a somewhat idealized case that for aresonant mode with frequency f₀=ω₀/2π, Q˜ω₀/Δω, where Δω is a spreadabout ω₀ (e.g., full width at half-maximum). Thus, Q may depend on theresonant frequency as well as on physical characteristics of the cavity.Determination of Q for Schumann resonances is much more complex than ina simplified ideal case, but it can be shown that for the lowestfrequencies, Q is quite low—in the single digits. While Q is higher forhigher frequency modes, these modes tend to overlap with one anothersuch that natural excitations of just one high-frequency mode by itselfare rare (if they occur at all), especially when the source is lighting,which is wideband in frequency and localized in space (and thereforeincoherent in both space and time). This explains why lightning tends toexcite only the first few Schumann modes. The low Q values, as well asthe incoherence of lightning events, explain why the energy carried inSchumann resonances is observed to dissipate fairly quickly.

In regard to low-loss eigenmodes of the Earth-ionosphere waveguide thatmay be suitable for efficient wireless transmission of power, Schumannresonances demonstrate the physical reality of the Earth-ionospherewaveguide cavity. But the dominant modes have values of Q too low to beuseful for efficient power transmission, since too much energy is lostdue to interactions in the ionosphere (ohmic losses). At higherfrequencies, where Q is also higher, the overlapping modes make itdifficult to excite just a single mode by itself. As a practical matter,Schumann modes are only known to be excited by lightning, so theirapplicability to engineered wireless power transmission may be limited.In any case, to the extent that a transmitter might couple some fractionof power to one or more Schumann modes, any resulting power transmissionwould not be efficient at least for the reasons above.

Whispering Gallery Mode.

Whispering gallery waves are waves that propagate along the innersurface of a sphere. Initially studied and described for sound waves,whispering gallery modes of VLF and low frequency radio frequency (RF)transmissions propagating along the underside of the ionosphere are alsoknown to occur. Analytically, these RF modes have been described interms of repeated reflections of waves at the underside of theionosphere, without consideration of the presence of the Earth itself.The resulting phenomenon is a surface wave guided by the ionosphere.However, just as Zenneck waves suffer ohmic losses to the outerwaveguide boundary of the ionosphere, whispering gallery waves suffersuch losses to the inner waveguide boundary of the Earth's surface. So,although whispering gallery modes again demonstrate that the volumebetween the Earth's surface and it ionosphere can act like a waveguidecavity, the proximity of these waves to the ionosphere and theterrestrial surfaces losses they suffer appear to limit the practicalityof wireless power transmission on a global scale, to the extent that atransmitter could couple any power to such modes.

Determining Eigenmodes of the Earth-Ionosphere Waveguide.

The modes and phenomena discussed above indicate that solutions ofMaxwell's Equations for the Earth-ionosphere waveguide can be plausiblyexpected to yield low-loss eigenmodes. At the same time, the known modessummarized above do not appear to be candidates for such eigenmodes. Asdiscussed above, the Zenneck wave modes indicate that the terrestrialsurface of the Earth can serve as the inner boundary to a globalwaveguide, but that the ionosphere cannot be neglected. Schumann andwhispering gallery modes indicate that the ionosphere can serve as theouter boundary of the global waveguide. However, the physicalinteractions of an electromagnetic wave with the plasma of theionosphere are complex, frequency dependent, and anisotropic.Accordingly, systems and methods disclosed herein involve bothanalytical simulations and empirical verification to predict, identify,and excite low-loss eigenmodes of the Earth-ionosphere waveguide. Theanalytical simulations and empirical verification form the analyticaland empirical components described above.

In accordance with example embodiments, simulations model theterrestrial surface of the Earth as the inner boundary of a sphericalwaveguide cavity and the ionosphere as the outer boundary. In thesimulations, existing models of the ionosphere, accounting for plasmaphysics, may be used. Various computational and numerical techniques maybe used to improve the speed and efficiency of the computationaloperations.

The simulations serve primarily two aspects of the analytic component ofthe disclosed approach. In accordance with example embodiments, thefirst aspect is determining possible eigenmodes, given variousparameters or parameter ranges of the model. Non-limiting examples ofparameters include Earth surface conductivities or conductivity maps,diurnal variations of ionization in the ionosphere, and atmosphericvariations of the ionosphere. By initially employing a somewhatsimplified model, the first aspect of simulation can help map out asolution space for one or more low-loss eigenmodes and/or identifyfamilies of low-loss eigenmodes according to expected or plausibleparameter ranges, without demanding potentially impractical computingresources.

In accordance with example embodiments, the second aspect of theanalytic component is predicting a degree or strength of couplingbetween the electromagnetic field of a simulated excitation source(e.g., transmitting antenna) and a given one of the possible eigenmodes.In an example embodiment, the predicted coupling strength can becomputed as an “overlap integral” that can be expressed as an integralover an inner product (dot product) of the eigenmode electromagneticvector field and the conjugate of the simulated electromagnetic vectorfield of the simulated excitation source. Equation 3 is an expressionfor a form of the overlap integral S,

$\begin{matrix}{{S = \frac{\int{E_{{Eigen}\mspace{14mu} {mode}} \cdot E_{Source}^{*}}}{\sqrt{\int{E_{{Eigen}\mspace{14mu} {mode}} \cdot E_{{Eigen}\mspace{14mu} {mode}}^{*}}}\sqrt{\int{E_{Source} \cdot E_{Source}^{*}}}}},} & \lbrack 3\rbrack\end{matrix}$

where E_(Eigenmode) is the field of the Eigenmode, E_(Source)* is thecomplex conjugate of the field of a simulated source, such as dipoleantenna, and the integral is taken over a geometric region of interest(e.g., over the volume of a waveguide). The multiplication is a dotproduct of the field vectors at each spatial point within the waveguide,and the integral is taken over the volume of the waveguide and somespecified time interval. In the normalized form shown, the overlapintegral yields a dimensionless value in a range from zero to one,corresponding to no coupling to complete coupling, respectively.

In further accordance with example embodiments, the empiricalverification entails coupling the electromagnetic field of an actualexcitation source (e.g., a transmitting antenna) to one or more of theeigenmodes predicted by the simulation(s), detecting and measuringwireless power received from the source via coupling to the eigenmode,verifying the received power as that of the excitation source,determining the strength of the coupling to the eigenmode, refining theanalytical description of the eigenmodes, and adjusting the excitationsource to enhance the coupling.

Taken together, the two aspects of the analytic component help guide theempirical verification by indicating how to empirically adapt wirelesspower transmissions to align with characteristics of predictedeigenmodes, and therefore enhance the likelihood and effectiveness ofcoupling. The analytic component also helps determine an expectedcoupling strength that can then be compared with measurements, and in sodoing provides a statistical confidence that the empirical measurementscorrespond to power carried in an eigenmode. The empirical measurementscan then be used to improve the analytical description of theeigenmodes, improve the predictions of coupling by using thebetter-described eigenmodes, and further adapt and adjust transmissionsto better couple to the eigenmodes.

Two examples of the analytic component as applied to Zenneck modesillustrate how simulation may be used to predict expected coupling andpower propagation, and also to further quantify the evaluation ofZenneck modes discussed above. In both example applications, a radialZenneck mode solution, such as that described above for a cylindricalsurface wave, takes the place of a predicted eigenmode in the overlapintegral. As such, the first aspect of the analytic component—i.e.,simulation to identify eigenmodes—is bypassed, using instead a derivedZenneck mode solution. Note that use of a Zenneck wave also implicitlyignores the ionosphere, since the mathematical solution is derived inthe absence of any boundary condition above the guiding surface.

For the first example, the simulated source of power transmission wastaken to be a 100 m tall helical antenna having a vertical axis and sixturns and transmitting at a frequency of approximately 20 kHz. A fullthree-dimensional (3D) simulation was used, but carried out over onlyapproximately six wavelengths in the radial direction owing to otherwiseexcessive computing resources that would have been needed. FIG. 5 is aconceptual illustration of the simulation configuration. The upper panelshows the simulation region, also indicating the location of perfectlymatched layers (PMLs) used as a boundary matching technique (see theexplanation below). The lower panel depicts a zoomed-in view of thehelical antenna and the surrounding volume. The computation yielded anoverlap of ˜34% (or S≈0.34). A similar result was obtained by replacingthe simulated helical antenna with a simulated 100 m tall verticaldipole antenna. Such a low level of overlap, which would likely beinadequate for wireless power transmission to be achieved efficiently ona large scale, may be due to the restricted distance range over whichthe simulation was carried out.

For the second example, the simulation was simplified by takingadvantage of cylindrical symmetry to carry out the computation in twodimensions (2D), which also allowed the use of simplified sources in thesimulation. For example, cylindrically symmetric sources, such aselectric or magnetic dipoles, can reasonably approximate a helicalsource. In this case, the simulation was carried out over a geometrycorresponding to a flat disk having a radius roughly equal to a meridiandistance on the globe from a pole to the equator of the Earth (about 10⁴km). The thickness of the disk was taken to correspond roughly to theheight of the ionosphere (about 100 km), although the upper boundary wasnot modeled as the ionosphere. Instead, it was treated as a sort ofartificial mathematical boundary that allowed the simulated solution tobe matched to a hypothetical continuation of the solution beyond theboundary. In an example embodiment, a technique of perfectly matchedlayers (PMLs) may be used for the boundary matching. As is known, a PMLis a mathematical technique used in numerical simulations involving waveproblems having open boundaries. A PML provides an artificial layermathematically constructed to “absorb” incident waves from an adjacentnon-PML region, without reflecting them back into that region.

FIG. 6 is a conceptual illustration of the simplified simulation. Thetop panel (a) on left of FIG. 6 represents the 2D geometry used in thesimulation, in this case just a rectangle representing an edge-on viewof disk. The length is equal to disk radius and the height is equal tothe layer thickness; the excitation source (e.g., a vertical dipole) istaken to be at the lower left corner of the rectangle, a point whichalso defines the origin of the plot. The bottom panel (b) on the leftshows a simulated electromagnetic field propagated from the simulatedsource, a dipole radiating at ˜20 kHz. The aspect ratio of the two leftpanels is true to the actual distances, so the disk as depicted is quitethin and the details of the simulated field may be difficult to discern.

The top panel (c) on the right side of FIG. 6 shows a zoomed-in view(magnified) of the simulated electromagnetic field, also indicating thelocation of the PML used in the simulation, and the bottom right panel(d) shows a zoomed-in view of the analytic solution of a Zenneck mode inthe absence of any excitation source. An apparent spatial periodicity ofboth fields, visualized as grayscales, can be seen in the magnifiedviews. The overlap integral is applied to the simulated field and theZenneck mode, and carried out over the geometry of the disk. Even whenincluding an extra loss factor of two to account for assumed hemisphericsymmetry, the result of the overlap computation yields an overlap ofgreater than 90% (or S≈0.9). It may therefore be concluded thatsignificant overlap or coupling may not be implausible when excitingZenneck modes, at least for an assumed flat geometry. However, when thecurvature of the Earth is incorporated into the simulation, thefeasibility of Zenneck modes for wireless power transmission on a globalscale diminishes significantly.

This can be seen in FIG. 7, which illustrates conceptually what happensin the second example when the simulation accounts for the curvedsurface of the Earth. For purposes of illustration, the origin of theplot is taken to be the North pole, so points 90° south line on theEquator. The location of the PML used in the simulation is alsoindicated. It will be appreciated that any point on a sphere can betaken as an origin and “local pole,” with angular and surface distancesmeasured with reference to this point. The same applies to otherillustrations of spherical geometries discussed below.

The illustration in FIG. 7 shows about 1,500 km of the Earth's surfaceand an overlying mathematical boundary at a height corresponding to theionosphere, though again, the simulation did not account for theionosphere. A vertical dipole excitation source is taken to be at thepole, which is also the origin of the plot as described above. Here, theamplitude of the simulated field (as represented by a grayscale) appearsto effectively vanish within about 1,000 km from the source, showingthat the simulated field radiates away its energy as it propagatesaround the curvature of the Earth. This traces out how a Zenneck wavebehaves on the Earth's spherical surface, with the ionosphere ignoredfor the moment. For the material properties of the Earth, the Zennecksolution produces a wave that decays exponentially in the verticaldirection with a scale height of ˜100 km, depending on the frequency ofthe wave. If the Earth's surface is assumed to be more lossy than itactually is, the Zenneck mode is more strongly constrained to propagatealong the curved surface. But then the surface losses becomesignificant. Zenneck modes can also be more strongly constrained topropagate along the Earth's curved surface if lower frequencies areconsidered, but in this case, the vertical scale height increases wellbeyond where the ionosphere begins. And though the ionosphere wasneglected in this simulation, it cannot be in actual practice. Thus,severe I²R losses in the ionosphere would then be expected characterizethe low-frequency, high-scale-height Zenneck modes.

The simulated field in FIG. 7 can also be interpreted as a sort of“probe” of possible natural modes of the Earth. For example, thebehavior noted above highlights what would be expected from excitingZenneck modes with a dipole source (and the ionosphere neglected), andreinforces the conclusion that doing so is not likely to achieveefficient wireless power transmission on a global scale. A probe fieldmay also (or instead) couple to a low-loss eigenmode, and simulation maysimilarly be used to evaluate both the eigenmode and the strength ofcoupling that could be expected with an actual source. In accordancewith example embodiments, this can be accomplished by applying the firstaspect of the analytical component, namely using simulation to determinesolutions of Maxwell's equations applied to the Earth-ionospherewaveguide. That is, determining low-loss eigenmodes with simulationsthat model both electrical properties of the Earth's surface and plasmaphysics of the ionosphere. As described above, analytical descriptionsof low-loss eigenmodes may then be used in the overlap integral ofEquation 3.

In an example embodiment, a simulation incorporates a model of theEarth's surface, including electrical properties, and a model of theionosphere, including electrical properties and plasma physics. Giventhis physical model of the Earth-ionosphere waveguide, the simulationnumerically searches for solutions to Maxwell's Equations, yielding oneor more eigenmodes. Simulation results are generated in the form anumerical representation of each eigenmode, and correspondingeigenvalues including propagation constants and complex valuedeigenfrequencies for each eigenmode. In particular, the eigenfrequencyfor a given eigenmode is given by λ=ω+iδ, where i=√{square root over(−1)}. In this expression, ω=2πf is the angular frequency and δ is adamping factor. Following reasoning similar to that described above, itcan be shown that the Q factor is given by Q=ω/2δ. The numericalrepresentation of an eigenmode may take the form of electric andmagnetic vector field components at discretized spatial points withinthe electric and magnetic vector field solution space. A simulation maybe started with initial estimates for eigenvalues, and may yield justone eigenmode or multiple eigenmodes.

FIG. 8 shows an example low-loss eigenmode identified by way of anumerical simulation applied to a spherical waveguide cavity model that,for purposes of illustration, includes simplified electrical propertiesof the Earth's surface and simplified material-physical properties ofthe ionosphere. Again, the location of the PML used in the simulation isindicated. By way of example, the simulation assumed a groundconductivity of 0.001 S/m, a single ionosphere conductivity of 10⁻⁶ S/m.An initial frequency estimate of 7.5 kHz provided a starting point forthe simulation. The simulation yielded an eigenmode having acomplex-valued frequency λ=(7.4954+0.02811i) kHz, from which a value ofQ≈133 was derived. The simulation was carried out in 2D over a meridianarc from the “north” pole (taken to be the origin) to the “south” pole(antipode), and models the ionosphere above as an overlying layer havinga lower boundary and an upper boundary and characterized between theupper and lower boundaries by a single conductivity.

The example eigenmode produced by the simulation is characterized bystanding waves corresponding to a resonant mode of the Earth-ionospherewaveguide. Because of the low frequency of ˜7.5 kHz, the ionosphere canact to trap the waves within the cavity. A semi-circle in FIG. 8represents the geometric region over which the example eigenmodesolution was simulated. The thinness of the layer as depicted in thefigure may make the form of the solution difficult to discern. However,in an approximately 1,000-km portion of the region starting at the poleshown in magnification, the spatial periodicity of the eigenmode—roughlyevery 40 km—becomes apparent. The amplitude of the eigenmode,represented in a grayscale, appears to diminish with distance from thepole. But unlike the Zenneck wave solutions where the diminishingamplitude represents energy dissipation due to radiation into spaceand/or ohmic losses, the decrease seen in the eigenmode is predominantlygeometric spreading or dilution. As the wave propagates past the equatortoward the antipode, the intensity increases again as the geometricspreading reverses.

Viewed from above the pole, the equator would appear as a circle and thestanding waves would appear as concentric rings diverging from theorigin (pole), in analogy to concentric surface ripples diverging from apebble dropped in water. Continuing with the analogy, if mechanicallosses (e.g., viscosity and friction) are ignored for the surfaceripples, the total energy in each ripple will remain roughly constant,but the energy density in each ripple decreases as the circumferenceincreases with distance from the center. This corresponds to thegeometric dilution of the eigenmode with distance from the pole to theequator. If the spreading surface ripples hit a concentric circularreflecting boundary, their motion will reverse and they will reconvergetoward the center. As they do, the energy density dilution will alsoreverse. This reconvergence corresponds to the eigenmode waves past theequator and toward the antipode.

The eigenmode shown by way of example in FIG. 8 is one of severalexamples yielded by somewhat simplified simulations that can be used toprovide general characteristics and behaviors of solutions as a functionof properties of the Earth-ionosphere waveguide. For example, someeigenmodes for models with low-conductivity ionospheres and/or with anextremely high-altitude ionosphere undersurfaces share similarities withsurface-hugging Zenneck modes. Solutions for models withhigh-conductivity ionospheres and a very low-conductivity Earth surfaceshare similarities with whispering gallery modes. Solutions for modelswith high-conductivity upper and lower “hard” boundaries sharecharacteristics with ionosphere-hugging Schumann modes. Althoughsolutions that resemble Zenneck waves, Schumann modes, and/or whisperinggallery modes lack characteristics making them suitable for global powerdistribution, they illustrate how even relatively simple configurationsof the physical model used in the simulations can begin to map out aneigenspace at a coarse level.

Simulations that incorporate more realistic—if still overlysimplistic—models of the ionosphere yield low eigenvalues having Qvalues on the order of ˜200 or greater. As such, these eigenmodes may beconsidered low-loss in the sense that relatively little of their energyis lost to the waveguide boundaries—i.e., the Earth's surface and theionosphere. Thus, a low-loss eigenmode may be subject to geometricdilution with distance from the origin (and geometric re-concentrationtowards the antipode), but only a relatively small amount of totalenergy is lost to the ionosphere or other loss processes. To the extentthat a source can couple at least a fraction of its power to such aneigenmode, that coupled fraction of power may be transmitted around theglobe with little loss.

This is shown in FIG. 9, which illustrates the field pattern of asimulated dipole emitting in the Earth-ionosphere waveguide, and wheresimulation is carried in just two dimensions out over the upperhemisphere of the geometry of FIG. 8. In order to extend both to thelower hemisphere and beyond the ionosphere, mathematical boundaries wereimposed at the equator and above of the ionosphere layer to match thesolutions within the solution volume with hypothetical continuations ofthe solutions beyond the boundaries. In an example embodiment, atechnique of perfectly matched layers (PMLs) may be used for theboundary matching.

The upper left panel (a) in FIG. 9 spans approximately the first 2,500km from the simulated source, and the inset panel (b) below zooms in onthe first ˜250 km from the source. As expected for dipole radiation, thepattern is roughly circular near the source, so that a large proportionof the energy near the source is directed toward the ionosphere.Consequently, within ˜200-300 km from the source, a large fraction ofthe initial power is lost to interactions between the field and thepartially ionized plasma. However, at increasing distance from thesource, the direction of energy propagation is predominantly parallel tothe Earth's surface and the ionosphere. This radiation pattern iscustomarily referred to as “broadside” with respect to the axis of thedipole source. Although the broadside portion of the field in thissimulation accounts for only a small fraction of the initial energy—mostof which is lost to the ionosphere—its power propagates nearly withoutany further loss across the globe and within the Earth-ionospherewaveguide. Thus, in the top left panel, the broadside radiation patterncan be seen to continue past the right edge. The apparent decrease inintensity with distance is due primarily to geometric dilution, asexplained above. However, ohmic losses in the ionosphere have largelyleveled off by about 1,000 km from the source.

This is shown by way of example in the right panel (c) of FIG. 9, whichis a plot of ohmic losses in the ionosphere with distance from thesource. While nearly about 99% of the initial power is dissipated in theionosphere by 1,000 km from the source, the remaining 1% remainsrelatively unchanged beyond this distance.

The simulation illustrated in FIG. 9 indicates that the simulated dipolefield has coupled with one or more low-loss Eigenmodes, although most ofits initial energy is lost to the ionosphere. Thus, while evidently notan efficient coupling source in terms of injected energy, the simulateddipole nevertheless acts as a probe of eigenmodes, demonstrating theirexistence—at least as predicted by simulation. The apparent inefficiencyof coupling in terms of injected power is not necessarily unexpected inthis example, and is more indicative of the prevalence ofionosphere-directed energy flow (and resultant losses) near the dipolesource than of any characteristic of low-loss eigenmodes or thefeasibility of constructing an excitation source better suited forefficient coupling.

Exciting and Detecting Eigenmodes of the Earth-Ionosphere Waveguide.

One approach to coupling power into an eigenmode at one location andreceiving some portion of the power at another location is to operate alow-frequency dipole transmitter at relatively high power and deploy oneor more receivers at remote locations. This approach is analogous to thesimulated dipole illustrated in FIG. 9, and may therefore be suboptimalin terms of coupling efficiency. However, by measuring the receivedpower and adjusting the transmitter properties based on themeasurements, transfer properties of the transmission path fromtransmitter to receiver could be determined empirically. Accordingly,this approach could be considered an empirical determination of atransfer function of the Earth-ionosphere waveguide. The transferfunction could then be used to inform and guide design and constructionof more optimal transmitter and receiver systems.

Alternative or additional approaches may take advantage of simulationsby using predicted properties of eigenmodes to introduce some degree ofoptimization from the start.

In accordance with example embodiments, properties of predictedeigenmodes may be used to infer characteristics of a simulated and/oractual excitation source that make coupling of the source field to theeigenmodes efficient. More particularly, simulation of an excitationsource characterized by significant broadside radiation indicatesincreases in the efficiency of coupling in the context of simulations.Computing the overlap integral for such a simulated source with one ormore predicted eigenmodes may then provide a parameter space of expectedpower level measurements that may be used to empirically validate thepredicted eigenmodes and help guide design and construction of actualexcitation sources.

The solution for the simulation of FIG. 8 illustrates just one eigenmodefor a simplified model of the Earth-ionosphere waveguide. In accordancewith example embodiments, multiple runs of the simulations may becarried out to cover a range of terrestrial (ground) conductivities andionospheric conductivities. Simulations may further be carried out in 3Dusing detailed models for the ionosphere that include detailed plasmaphysics to account for anisotropic and inhomogeneous conductivities, aswell as diurnal variations (e.g., daytime/nighttime variations). Fromthese multiple simulation runs, an eigenspace may be mapped outrepresenting eigenmodes that sample basis vectors of solutions toMaxwell's Equations for the Earth-ionosphere waveguide. Simulations ofsources may then be used to determine physical characteristics ofemitted electromagnetic fields that most efficiently couple with one ormore low-loss eigenmodes among those identified from the eigenmodesimulations.

For example, the source simulation represented in FIG. 9 suggests thatreplacing the simulated dipole with a more broadside emitter would leadto better coupling. Also, the source frequency needs to be low enough sothat waves are largely trapped (e.g., reflected) by the ionosphere. Inaccordance with example embodiments, this type of analysis may beexpanded with a set of low-loss eigenmodes from simulations to determinesource configurations that yield large predicted overlap integrals. Thedetermined source configurations can also be translated into desirableelectrical properties of actual sources for deployment in empiricaldetermination of coupling. In practice, any source may couple tomultiple eigenmodes at differing levels of efficiency. As such, theoverlap integral may represent the total amount of source coupling tothe multiple eigenmodes, with any uncoupled portion, R=S−1, representinglosses and/or coupling to eigenmodes that are not included in theoverlap integral. Selectively omitting particular eigenmodes from thesimulated overlap integral may allow individual contributions (i.e.,coupling strengths) to be deduced. By further accounting for propertiesof omitted eigenmodes, such as possible spatial rates of decay, theireffect on overall coupling modalities may be determined. This can beused to further explore and design desirable properties of actual(empirical) sources.

FIGS. 10A, 10B, and 10C conceptually illustrate “flattening” of anelectric field, such as dipole field. A radiation pattern of the formproduced by a vertical dipole, such as a dipole antenna, is shown inFIG. 10A. In line with the discussion of FIG. 9, a high proportion ofthe field vectors have large vertical components, so that the overallvector field similarly has a substantial upward component. As described,the relatively large proportion of vertical vector field componentsleads to substantial I²R losses in the ionosphere (and somewhat less soin the Earth's surface). FIG. 10B shows a slightly flattened version ofthe dipole field. In this case, the field vectors have an increasedamount of horizontal (broadside) components. Consequently, there is lessloss to the ionosphere and more energy that can couple to a low-losseigenmode. FIG. 10C shows still more flattening, and thus an even betterprospect for coupling to a low-loss eigenmode. The simplified depictionsin FIGS. 10A, 10B, and 10C illustrate conceptually how characteristicsof an empirical source might be adapted to enhance coupling to an actuallow-loss eigenmode. As described later, a directed, flattened fieldpattern may be achieved using an array of antennas. Some alternativephysical array configurations are first considered, leaving asidetheoretical details of how field flattening (and other forms of shaping)may be achieved in practice.

In accordance with example embodiments, an empirical source deploymentmay take the form of a multiplicity of commonly located RF transmitterelements configured to operate at tunable frequencies, tunableamplitudes, and tunable relative phases so as to generate a directedradiation pattern at one or more specified frequencies. In particular,the tunable amplitudes and relative phases may be adjusted so that, inconcert, the multiplicity produces a substantially flattened radiationpattern with respect of a vertical axis and having power directedradially outward from the common location. The flattened radiationpattern may be parallel to a local horizon such that little to no poweris radiated toward the ionosphere or into the ground. Further, theamplitude and phase adjustments may be made such that the radiationpattern is isotropic with respect to an azimuthal angle measured aboutthe vertical axis. Alternatively, the radiation pattern may be arrangedto have phase and/or amplitude be a function of the azimuthal angle.

In an example embodiment, each RF transmitter element may include awaveguide-coupling element, such as an antenna, and an electricallyconnected driving circuit. Each transmitter element may have its owndriving circuit, or some or all of the transmitter elements may beconnected to one or more common driving circuits, where each connectionallows the amplitude and phase of each transmitter element to beindividually controlled. In an example embodiment, the antenna may be anelectric dipole antenna or a helical (magnetic dipole) antenna. Otherconfigurations of RF transmitter element are possible as well.

In accordance with example embodiments, the multiplicity of RFtransmitter elements at a common location may take the form of an arrayof electrically connected RF transmitter elements distributed over anarea. Illustrative examples of arrays of antennas are shown in FIG. 11.Panel (a) is an illustration of a spatial (e.g. rectangular) array ofdipole antennas; panel (b) is an illustration of spatial (e.g.rectangular) array of helical (magnetic dipole) antennas; panel (c) isan illustration of a vertical (stacked) array of helical antennas. Otherconfigurations could include both types of antennas and/or other typesof transmitter elements. In an example, the area may be roughly the sizeof a football field or similarly sized footprint.

The electrical connections to and among the transmitter elements of thearray may include controllable elements that enable the frequency,amplitude, and phase of transmitter elements to be dynamically adjusted.In an example, the controllable elements may include adjustable activeand/or passive circuit components. The frequency may be adjusted to bewithin a range of expected eigenfrequencies of low-loss eigenmodes.Further, the phase adjustments may enable the relative phases of some orall pairs of transmitter elements to be adjusted according to specifiedamounts. In accordance with example embodiments, amplitudes and relativephases of transmitter elements may be adjusted so as to generate asubstantially broadside radiation pattern. As discussed above, such aradiation pattern can enhance coupling efficiency to low-loss eigenmodesby suppressing radiation directed toward the ionosphere andconcentrating it in a broadside direction where it may couple with oneor more eigenmodes.

Also in accordance with example embodiments, by making appropriateadjustments, the array of transmitter elements may present an electricalaperture significantly larger than its physical size. In particular, theaperture size may be made large enough in comparison to the range ofwavelengths corresponding to the range of eigenfrequencies so that asignificant portion of the emitted power may be concentrated within thenear field of the eigenmodes. In this way, non-radiative, near-fieldcoupling with one or more eigenmodes may be achieved.

In accordance with example embodiments, an operational frequency rangeof 5-100 kHz may be used, corresponding to a wavelength range ofapproximately 3-60 km. Transmissions in this range can generally beexpected to reflect off the ionosphere, particularly if thetransmissions are predominantly broadside with respect to the verticaldirection. Also within this frequency range, an array of electricallyconnected transmitter elements within an area characterized by afootball field or similar facility will be well within the near field ofRF transmissions. As such, relatively precise control of phases andamplitudes of the transmitter elements will typically be needed tocontain the power against reactive electromagnetic forces among thetransmitter elements. In the given example frequency range, the timingrequirements of electrical control elements for achieving this precisecontrol can generally be met with existing and/or available hardware andsoftware devices for implementing the electrical control elements. Forexample switching speeds of active circuit elements such as transistorsand integrated circuits typically well exceed timing corresponding tothe example operational range.

In accordance with example embodiments, adjustments of electricalconnections to and among the transmitter elements may be controlled byone or more computing systems, such as those discussed above. Suchsystems may include one or more processors or controllers, and memoryfor storing instructions that when executed by the one or moreprocessors carry out operations including control of the electricalconnections so as to achieve target and/or desired frequencies,amplitudes, and phases of the transmitter elements. The operations mayalso include computations for determining frequencies, amplitudes, andphases of the transmitter elements given one or more sets of operationalcriteria. For example, operational criteria could specify one or moreeigenmodes to which coupling is desired. The one or more computingsystems may also include a user interface for user control andmonitoring. Further, the one or more computing systems may be connectedto a communications network providing a communicative connection to oneor more other computing systems, including one or more that controlanother, remotely located array of transmitter elements, and/or one ormore that control one or more remotely located wireless power receivingdevices.

As discussed above, the conductivity of the ionosphere is generallyinhomogeneous and anisotropic. In addition the height of the D layer—thelowest layer—above the terrestrial surface varies over the course a day,being lowest during daytime, when direct solar energy increases thelevel ionization, and highest during nighttime when the level ofionization decreases in the absence of direct solar energy. As a result,the conductivity of the ionosphere is typically non-uniform andtime-dependent. In accordance with example embodiments, dynamic controlof frequencies, amplitudes, and phases of the transmitter elements ofthe array may be used compensate for non-uniformity and time-dependenceof the ionosphere. For example, during times when the D layer is lowest(i.e., daytime), adjustments may be made to generate a more flattenedradiation pattern so as to avoid directing power into the D layer.

In practice, power transmitted with a system or apparatus such as theexample array of transmitter elements may couple to more than oneeigenmode, and even at some level to other possible modes, such as someof those discussed above. This may be particularly the case for initialempirical trials at a stage when the eigenmodes and eigenvalues haveonly been identified by simulations. Accordingly, a receiver device maybe configured for coupling power in from one or more eigenmodes.

In accordance with example embodiments, one or more receiver devices orsystems may be deployed at one or more locations remote from atransmitter system in order to detect and measure transmitted power. Byconfirming that the detected power is from the transmitter system, andby measuring the strength of the detected power, properties of thereceived power may be evaluated to determine the strength of coupling ofthe transmitted power to one or more eigenmodes, and to further refinequantitative descriptions of the eigenmodes beyond what is provided bythe simulations alone. Properties of the detection and the refinedquantitative descriptions may then be used to further adjust thetransmitter system to better couple to the one or more eigenmodes.

In accordance with example embodiments, a receiver device may include areception element, such as an antenna, and an electrically connectedreceiving circuit. The receiving circuit may include adjustable elementsfor tuning to a range of frequencies. In an example embodiment, theantenna may be an electric dipole antenna or a helical (magnetic dipole)antenna. Other configurations of a receiver device are possible as well.

In accordance with example embodiments, a receiver device can be used ina standalone mode, or be one element in an array of electricallyconnected receiver elements distributed over an area. An illustrativeexample of an array of receiving antennas is shown in FIG. 12. Panel (a)is an illustration of a spatial (e.g. rectangular) array of dipoleantennas; panel (b) is an illustration of spatial (e.g. rectangular)array of helical (magnetic dipole) antennas; panel (c) is anillustration of a vertical (stacked) array of helical antennas. Otherconfigurations could include both types of antennas and/or other typesof radiating elements. In an example, the area may be roughly the sizeof a football field or similarly sized footprint.

In an example embodiment, a receiver device may measure 3D vectorcomponents of electric field and magnetic field components of thetransmitted electromagnetic field. The vector field components mayfurther be measured as a function of time. For example, a receiverdevice measure E(t)={E_(x)(t), E_(y)(t), E_(z)(t)} and H(t)={H_(x)(t),H_(y)(t), H_(z)(t)}, where x, y, and z are local rectangular coordinatesand t is time. Other coordinate systems could be used as well, such asspherical coordinates. Measurements of E(t) and H(t) over time may thenbe used compute received power at the location of the receiver device.

In an example embodiment, power transmitted by a transmitter apparatus,such as an array or transmitter elements, may be encoded withinformation such that a receiver system or apparatus may affirmativelyidentify the received power as according to the encoded information. Forexample, the transmitted power may take the form of a known pulsesequence that a receiver device may recognize. Other forms ofinformation encoding could be used as well. Upon detecting power andidentifying the encoded information, the receiver device can thenconfirm that the received power originates from the transmittingapparatus. As such, measurements of the received power may then be usedto evaluate the strength of coupling to one or more eigenmodes.

In accordance with example embodiments, simulations may be used topredict an overlap integral between a simulation of the transmittingapparatus and one or more predicted low-loss eigenmodes. As described,the overlap integral is taken over a specified time duration and overthe entire volume of the Earth-ionosphere waveguide. Measurements ofdetected power by a receiver device represent just a portion of thecomputed overlap integral, and possibly just a portion of the specifiedtime duration. As such, measurements from one receiver device may beconsidered a sample point of an actual overlap integral, and thereforecorrespond to an analytically derivable fraction of the predictedoverlap integral.

In accordance with example embodiments, a multiplicity of receiverdevices may be deployed at multiple, different remote locations from atransmitter apparatus. A conceptual illustration of multiple,distributed receiver devices is shown in FIG. 13. Measurements ofreceived power at these receiver devices may then be summed together toform a partial discrete overlap summation that can be compared with aportion of the overlap integral. That is, S is approximated by a finitesum of discrete dot products that represents a fraction of the totalvolume integral. The partial sum of measured points may then be comparedwith a corresponding proportion or fraction of the computed overlapintegral from simulations. In carrying out the partial discretesummation, adjustments may be made for a relative time delays betweenthe transmitter apparatus and the receiving devices. Computationally,this adjustment may be achieved by taking a Fourier transform of localmeasurements of E(t) and H(t) and then carrying out the partial discretesummation in frequency space.

In accordance with example embodiments, simulations may also providepredicted values, or ranges of values, of E(t) and H(t) at one or moreremote locations relative to the transmitter apparatus. Actualmeasurements of E(t) and H(t) by a receiver device as one or more of theremote locations may then be compared with the predicted values.Analysis of measured values may thereby provide confidence estimates forconfirming that received power was carried by one or more eigenmodes.

By deploying multiple receiver devices remotely from a transmitterapparatus, a spatial mapping of E(t) and H(t) may be determined andcompared with predicted field values from the transmitter apparatus. Inthis way, receiver measurements may be analyzed to provide statisticalconfidence estimates for confirming coupling of the transmitterapparatus with one or more eigenmodes. Further, the parameters of thesimulation-based eigenmodes in the discretized computation of S may beadjusted to obtain a best fit to the spatially diverse measurements ofE(t) and H(t), for example in a least-squares sense. Doing so canimprove the physical description of the eigenmodes and further improvethe accuracy of the derived overlap integral.

In practice, initial measurements—those of eigenmodes identified largelyor only by simulations—may provide relatively low-confidence estimatesof coupling. However, such measurements may be used to refine analyticdescriptions of the one or more eigenmodes. The refined descriptionscould then be used determine adjustments to the transmitter apparatus soas to better (e.g., more efficiently) couple to the eigenmodes. Furthermeasurements by receiver devices could then provide increased confidenceof coupling confirmation. This process could be repeated iteratively toimprove operational capability of the transmitter apparatus in couplingwith one or more eigenmodes.

In accordance with example embodiments, an iterative process may also beapplied to empirically detecting eigenmodes without necessarily refiningtheir analytic descriptions. In this approach, initial simulation-basedestimates may again be used for initial detection of eigenmodes. Butinstead of refining analytic descriptions, iterative measurements may beused to determine an effective response function of the propagation pathfrom the transmitter apparatus to the receiver devices. The responsefunction may then be applied as a sort of “black box” system, such thatproperties of the transmitter apparatus and receiver device may beadjusted for efficient wireless power transmission, without necessarilyderiving a detailed analytical description of the underlying eigenmodes.

In accordance with example embodiments, an iterative process may bedeveloped in the form of a real-time feedback loop between thetransmitter apparatus and one or more receiver devices. In particular,real-time feedback may allow dynamic physical properties of theEarth-ionosphere waveguide to be incorporated into operation oftransmitter apparatuses and receiver devices. For example, operationalaspects of transmitters and receivers could be adapted to diurnalchanges in the ionosphere. In an example embodiment, ancillarymeasurements may be made with an ionosonde device configured, as isknown, for probing the height of the ionosphere as a function offrequency. The ionosonde device could be incorporated in a transmitterapparatus and/or in a receiver device, or could be a separate device.Measurements from the ionosonde could be used to determine real-timeconditions of the ionosphere, which could be made part of a feedbackloop for controlling the transmitter apparatus and/or one or morereceiver devices.

In accordance with example embodiments, a receiver device may take theform or a portable, deployable device that may be relocated to a varietyof locations with respect to a transmitter apparatus. Multiple portabledevices may be deployed such that their locations with respect to atransmitter apparatus may be adapted and reconfigured according topredicted or expected spatial properties of one or more eigenmodes. Forexample, an eigenmode determined from simulation may have predicted wavenulls and/or wave peaks at particular locations. While such predictionscould have relatively large uncertainties associated with the accuracyof one or more locations, mobile deployment of multiple portable devicesmay enable empirical spatial properties of standing waves of eigenmodesto be mapped out.

In an example embodiment, a portable receiver device could be smallenough to be hand-held. In another example, portable receiver devicescould be deployed using one or more types of mobile vehicles.Non-limiting examples of mobile vehicles for deployment of portablereceiver devices includes autonomous, semi-autonomous, and/orhuman-operated terrestrial and/or aerial vehicles. Aerial vehicles couldinclude airplanes, drones, and balloons. Terrestrial vehicles couldinclude cars, trucks, and trains. Other examples are possible as well.

FIGS. 14A and 14B are flowcharts illustrating example methods forpredicting and detecting eigenmodes of the Earth-ionosphere waveguide.In accordance with example embodiments, the methods can be implementedusing one or more transmitter apparatuses or devices, one or morereceiver apparatuses or devices, and one or more computing systemsconfigured for executing instructions for carrying out various steps andfunctions described herein. The computing system 100 illustrated in FIG.1 is an example of a computing system that could carry out steps andfunctions of the example methods. Instructions for execution by one ormore processors of the computing system could be stored as software,hardware, and/or firmware in a non-transitory computer-readable medium.Thus, steps, operations, and/or functions described herein may becarried out by the computing system when the instructions are executedby one or more processors of the one or more computing systems.

The method illustrated in FIG. 14A entails determining a strength ofcoupling between a transmitted electromagnetic wave and one or moreeigenmodes of the Earth-ionosphere waveguide. At step 1402, atransmitter apparatus transmits electrical power into a sphericalwaveguide bounded by a terrestrial surface and an ionospheric layer,wherein the electrical power is transmitted in an electromagnetic wave.In an example embodiment, the terrestrial surface is the surface of theEarth and the ionospheric layer is the ionospheric layer of the Earth,and the spherical waveguide is the Earth-ionosphere waveguide. Asdescribed earlier, the waveguide formed by the Earth and its ionosphereis not precisely spherical. As such, it may be understood that the term“spherical waveguide” applied to the Earth-ionosphere waveguide may betaken to mean approximately spherical. That is, there should be noambiguity to the term “approximately spherical” as it applies to theEarth-ionosphere waveguide.

At step 1404, one or more eigenmodes of the spherical waveguide arecomputed based on a mathematical model of the spherical waveguide thatincorporates electrical properties of the terrestrial surface and plasmaphysics of the ionospheric layer.

At step 1406, the transmitted electromagnetic wave is detected by areceiver apparatus remote from the transmitter apparatus.

Finally, at step 1408, a strength of coupling is determined between thetransmitted electromagnetic wave and the one or more eigenmodes by bymeasuring an amount of power received by the receiver apparatus in thedetected electromagnetic wave. In accordance with example embodiments,the determined strength of coupling may be used to calculated astatistical confidence that the detected electromagnetic wave wascoupled to the one or more eigenmodes during a time interval in which itwas detected by the receiver apparatus.

In accordance with example embodiments, computing the one or moreeigenmodes of the spherical waveguide (in step 1404) may entailnumerically solving Maxwell's Equations applied to the mathematicalmodel of the spherical waveguide and determining the one or moreeigenmodes from the numerical solution in the form of computed electricand magnetic field vectors at discrete spatial points of themathematical model of the spherical waveguide. In addition,eigenfrequencies and propagation constants for the one or moreeigenmodes may also be determined from the numerical solution.Numerically solving Maxwell's Equations applied to the mathematicalmodel of the spherical waveguide may also entail computing a numericalsimulation in two spatial dimensions and/or three spatial dimensions.

In further accordance with example embodiments, transmitting theelectrical power into the spherical waveguide (in step 1402) may entailtransmitting the electrical power at one or more of theeigenfrequencies. In an example embodiment, eigenfrequencies fromsimulations may be in a range of 5-100 kHz. It should be understood thateigenfrequencies outside of this example range are also possible, as areone or more other ranges of eigenfrequencies.

In accordance with example embodiments, transmitting the electricalpower into the spherical waveguide (in step 1402) may entail encodingparticular information into the transmitted electromagnetic wave.Detecting the transmitted electromagnetic wave by the receiver apparatusmay then entail detecting the encoded particular information. In furtheraccordance with example embodiments, the particular information could bea timing signature. For example, a timing signature could be, orinclude, one or more particular pulses or pulse sequences. Other typesencoding could be used as well. Non-limiting examples include amplitudemodulation and/or phase modulation.

In accordance with example embodiments, determining the strength ofcoupling between the transmitted electromagnetic wave and the one ormore eigenmodes (in step 1408) may entail measuring electric andmagnetic field vectors of the detected electromagnetic wave, and thencomparing the measured electric and magnetic field vectors with computedelectric and magnetic field vectors of the one or more eigenmodes in themathematical model of the spherical waveguide. The computed field valueswould be determined at a spatial location in the mathematical model thatcorresponds (at least as closely as possible within the localizationsupported in the model) to the location of the receiver apparatus.

In accordance with example embodiments, the one or more eigenmodes mayform standing waves of an electric and magnetic vector field. Further,the receiver apparatus may be configured as plurality of receiverdevices, each at a different remote location from the transmitterapparatus. Determining the strength of coupling between the transmittedelectromagnetic wave and the one or more eigenmodes (in step 1408) maythen entail predicting relative strengths of coupling among theplurality of receiver devices at each of the different remote locationsbased on standing waves, and comparing relative power measurements amongthe plurality of receiver devices with the predicted relative strengthsof coupling.

In further accordance with example embodiments, predicting relativestrengths of coupling among the plurality of receiver devices at each ofthe different remote locations based on standing waves may entailcomputing a numerical simulation of the transmitted electromagnetic wavebased on a mathematical model of transmission properties of thetransmitter apparatus. A discretized overlap integral may be computedbetween the one or more eigenmodes and the simulated transmittedelectromagnetic wave at simulated locations corresponding to thedifferent remote locations of the plurality of receivers. Thediscretized overlap integral can then be compared with an empiricaloverlap integral derived from measurements of electric and magneticfield vectors at the plurality of receiver devices.

In accordance with example embodiments, the transmitter apparatus may beconfigured as a plurality of electrically connected waveguide couplingelements. With this configuration, transmitting electrical power intothe spherical waveguide by the transmitter apparatus (in step 1402) mayentail transmitting electrical power by two or more waveguide couplingelements of the plurality while maintaining relative phases between thetwo or more waveguide coupling elements. The example method of FIG. 14Amay then further entail predicting a relative change in the couplingstrength between the transmitted electromagnetic wave and the one ormore eigenmodes that would result from one or more given changes in therelative phases between the two or more waveguide coupling elements, andalso entail predicting a change in the amount of power received by thereceiver apparatus in the detected electromagnetic wave based on thepredicted relative change in coupling strength. The relative phasesbetween the two or more waveguide coupling elements may then be adjustedby the one or more given changes in the relative phases duringtransmission. A determination could thus be made as to whether or not ameasured change in power received at the receiver apparatus is within athreshold of the predicted change in received power.

The method illustrated in FIG. 14B involves using measured power andinformation about computed eigenmodes to adjust properties oftransmitted power. At step 1422, a transmitter apparatus transmitselectrical power into a spherical waveguide bounded by a terrestrialsurface and an ionospheric layer, wherein the electrical power istransmitted in an electromagnetic wave.

At step 1424, one or more eigenmodes of the spherical waveguide arecomputed based on a mathematical model that incorporates electricalproperties of the terrestrial surface and plasma physics of theionospheric layer. Again, in an example embodiment, the terrestrialsurface is the surface of the Earth and the ionospheric layer is theionospheric layer of the Earth, and the spherical waveguide is theEarth-ionosphere waveguide.

At step 1426, a receiver apparatus measures at least a portion of powerfrom the transmitted electrical power.

At step 1428, an amplitude and phase of the electrical power transmittedby the transmitter apparatus are adjusted based on the computed one ormore eigenmodes and the measured power, so as to cause a predictedchange in the measured power.

Finally, at step 1430, a determination is made whether a measured changein power detected at the receiver apparatus is within a threshold of thepredicted change in measured power.

In accordance with example embodiments, computing the one or moreeigenmodes of the spherical waveguide (in step 1424) may entailnumerically solving Maxwell's Equations applied to the mathematicalmodel of the spherical waveguide and determining the one or moreeigenmodes from the numerical solution, as well as eigenfrequencies andpropagation constants for the one or more eigenmodes. Numericallysolving Maxwell's Equations may be done by computing a numericalsimulation in two spatial dimensions and/or three spatial dimensions.Transmitting could be done at one or more of the eigenfrequencies.Again, non-limiting example eigenfrequencies from simulations may be ina range of 5-100 kHz.

In accordance with example embodiments, the transmitter apparatus may beconfigured as a plurality of electrically connected waveguide couplingelements. With this configuration, transmitting electrical power intothe spherical waveguide by the transmitter apparatus (in step 1422) mayentail transmitting electrical power by two or more waveguide couplingelements of the plurality while maintaining relative phases between thetwo or more waveguide coupling elements. With this arrangement,adjusting the at least one of the frequency, amplitude, or phase of theelectrical power transmitted by the transmitter apparatus (in step 1428)could entail predicting a relative change in the coupling strengthbetween the transmitted electromagnetic wave and the one or moreeigenmodes that would result from one or more given changes in therelative phases between the two or more waveguide coupling elements, andalso entail predicting a change in the amount of power received by thereceiver apparatus in the detected electromagnetic wave based on thepredicted relative change in coupling strength. The relative phasesbetween the two or more waveguide coupling elements may then be adjustedby the one or more given changes in the relative phases duringtransmission.

In further accordance with example embodiments, the one or moreeigenmodes may form standing waves of an electric and magnetic vectorfield. Predicting the change in the amount of power received by thereceiver apparatus in the detected electromagnetic wave based on thepredicted relative change in coupling strength could then entailpredicting relative changes in the electric and magnetic field vectorsof the standing waves at the receiver apparatus.

In further accordance with example embodiments, the receiver apparatusmay be configured as a plurality of receiver devices, each at adifferent remote location from the transmitter apparatus. With thisarrangement, determining whether or not the measured change in powerreceived at the receiver apparatus is within a threshold of thepredicted change in received power (in step 1430) could entail computinga first empirical overlap integral derived from measurements of electricand magnetic field vectors at the plurality of receiver devices prior toadjusting the relative phases between the two or more waveguide couplingelements. A second empirical overlap integral could also be computedbased on measurements of electric and magnetic field vectors at theplurality of receiver devices after adjusting the relative phasesbetween the two or more waveguide coupling elements. A ratio of thefirst and second empirical overlap integrals could then be compared withthe predicted relative change in coupling strength between thetransmitted electromagnetic wave and the one or more eigenmodes.

III. Example Systems and Methods for Coupling to Low-Loss Eigenmodes ofa Waveguide

The discussion so far has focused on example systems and methods fordetermining, detecting, and validating properties of low-loss eigenmodesof the Earth—ionosphere waveguide. In accordance with exampleembodiments, the principles and techniques described above may beextended and expanded to enable low-loss eigenmodes to be usedpractically for global signal transmission, and in particular wirelesspower distribution on a global scale. In line with the disclosure above,a system for wireless power distribution needs to include a structurethat can launch or generate electromagnetic fields that can couple tothe one or more low-loss eigenmodes. Additionally, the system mustinclude a structure that can receive power carried by theelectromagnetic fields. In some examples, an identical structure canoperate either as a transmitter or receiver. In the context ofwaveguides, a structure that can couple to a waveguide (either as atransmitter or receiver) can be referred to as a waveguide coupler.

Accordingly, a waveguide coupler is disclosed that can excite low-losseigenmodes of the Earth-ionosphere waveguide. To do so, the waveguidecoupler can launch an electromagnetic excitation that overlaps (i.e.,mode-matches) one or more low-low loss eigenmodes of theEarth-ionosphere waveguide. Note that although the waveguide couplerdisclosed herein is described in the context of coupling to theEarth-ionosphere waveguide, the disclosed waveguide coupler can also beused to couple to other types of waveguides (e.g., waveguides withdifferent shapes, different boundary materials, and/or differentdielectric materials than the Earth-ionosphere waveguide).

In accordance with example embodiments, the waveguide coupler may beconfigured excite low-loss eigenmodes of a waveguide by incorporatingdesign characteristics based on properties of the waveguide. By way ofexample, the characteristics of a waveguide that can couple to theEarth-ionosphere waveguide can depend on properties of theEarth-ionosphere waveguide such as (i) properties of the ionosphericlayer, (ii) properties of the terrestrial layer, and (iii) properties ofthe eigenmode solutions of the Earth-ionosphere waveguide, as well asproperties of other known modes, such as those discussed above.

Based on these properties of the Earth-ionosphere waveguide, it can bedetermined that desired characteristics of a waveguide coupler that canexcite low-loss eigenmodes of the Earth-ionosphere waveguide includethat the coupler: (i) can generate an excitation that does not result inradiative losses outside of the ionosphere by penetrating through theionosphere or by being overly attenuated by it; (ii) can generate anelectromagnetic excitation that can overlap with at least one low-losseigenmode of the Earth-ionosphere waveguide; (iii) has a Q value that isat least as great as a threshold; and (iv) can operate without a compleximpedance matching network (in part because a complex impedance matchingnetwork can be lossy and can diminish otherwise high efficiency)

FIG. 15 illustrates a waveguide coupler 1500 that can excite low-losseigenmodes of the Earth-ionosphere waveguide, according to an exampleembodiment. As illustrated in FIG. 15, the waveguide coupler 1500 can bea vertically oriented helical coupler that includes a plurality of turnsof coil. FIG. 15 also illustrates parameters of the helical coupler 1500including (i) a height 1502 of each turn of coil (also referred to as apitch □ of the helical coupler 1500), (ii) a wire diameter 1504 (alsolabelled as diameter “d₀” in FIG. 15) of each turn of coil, (iii) awinding diameter 1506 (also labelled as diameter “d” in FIG. 15), and(iv) a height 1508 (also labelled as height “h” in FIG. 15) of thehelical coil coupler 1500. As discussed below, values for the parametersof the helical coupler 1500 can be determined such that the propertiesof the helical coupler 1500 satisfy the desired waveguide couplercharacteristics described above, including the ability to excite one ormore eigenmodes of the Earth-ionosphere waveguide.

In an embodiment, one or more of the desired waveguide couplercharacteristics can be achieved by designing the helical coupler 1500such that the coupler can achieve “supergain.” Supergain describes aphenomenon in which a radiating array of N elements can achieveincreased gain in comparison to a non-supergain array of N elements. Ina supergain array, the gain can grow linearly with the number ofantennas N, and can approach a magnitude of N² as the separationdistance between the antennas approaches 0. A gain of N² is asignificant increase in comparison to the maximum possible gain, N, forisotropic radiators spaced a half wavelength apart (i.e., anon-supergain array).

In an embodiment, achieving supergain can help the helical coupler 1500excite low-loss eigenmodes of the Earth-ionosphere waveguide. Asexplained above, some eigenmodes of the Earth-ionosphere waveguide canextend into the ionosphere, and thus an electromagnetic excitation thatoverlaps such an eigenmode can be attenuated by the ionosphere. However,by achieving supergain (e.g., increased directivity), the helicalcoupler 1500 can decrease the attenuation by generating anelectromagnetic excitation that has a suppressed vertical component andincreased horizontal directivity. The vertically suppressedelectromagnetic excitation can avoid interaction with the ionosphere,and therefore can decrease the attenuation by the ionosphere.

Returning again to FIGS. 10A, 10B, and 10C, each depicts athree-dimensional (3D) radiation pattern, according to an exampleembodiment. These figures can help illustrate the utility of supergainwaveguide couplers by showing the effect that achieving supergain canhave on a radiation pattern of a waveguide coupler.

FIG. 10A illustrates a radiation pattern 1000 of a radiating array 1002,which for the sake of simplicity, is depicted as a rod in FIG. 10A. FIG.10A also illustrates vectors, such as vector 1004, that represent thefields that result in the radiation pattern 1000. In this example, theelements of the radiating array 1002 are not closely spaced, andtherefore the array 1002 does not achieve supergain. In FIG. 10B, theelements of the radiating array 1002 are brought closer together. Asillustrated, bringing the elements closer together compresses theradiation pattern 1000. Additionally, the field vectors are morehorizontally directed than the field vectors in FIG. 3A. In FIG. 10C,the elements of the array 1002 are brought even closer together suchthat the array 1002 achieves supergain. As illustrated, the achievingsupergain further compresses the radiation pattern 1000. Additionally,the field vectors are more horizontally directed than the field vectorsin FIG. 10B.

FIGS. 16A and 16B each depict a polar radiation plot, according toexample embodiments. Like FIGS. 10A-10C, FIGS. 16A and 16B figures canhelp illustrate the utility of supergain waveguide couplers by showingthe effect that achieving supergain can have on a radiation pattern of awaveguide coupler. FIG. 16A illustrates a polar radiation plot 1600 of anon-supergain waveguide coupler, and FIG. 16B illustrates a polarradiation plot 1610 of a supergain waveguide coupler. Both figuresillustrate the polar radiation plot in complete 360 degrees in theazimuth plane. As illustrated in FIG. 16A, the radiation plot 1600includes a main lobe 1602, and the radiation plot 1610 includes a mainlobe 1612. As shown by comparing the main lobe 1602 and the main lobe1612, the radiation pattern 1610 of the supergain antenna has a largergain and more directivity than the radiation pattern 1602 of thenon-supergain antenna.

In an embodiment, supergain can be achieved by arranging the elements inclose proximity to one another. It may be counterintuitive to do sosince bringing the elements of the array close together can reduce thearray's gain due to an increase in mutual coupling between the elements.However, according to supergain antenna theory, the gain of the arraycan actually be increased by controlling a phase and/or magnitude of theradiation generated by each element.

In an embodiment, to apply supergain theory to the helical coupler 1500,the helical coupler 1500 can be modeled as an array where each turn ofcoil represents a radiating element. The helical coupler 1500 can alsobe viewed as equivalent to a stacked array of individual electricdipoles with magnetic dipoles disposed between the electric dipoles. Inthis representation, the spacing between the dipoles is equal to theturn-to-turn spacing of the helical coupler 1500.

In embodiment, the helical coupler 1500 can achieve supergain byarranging the turns of the helical coupler 1500 at a threshold distancewith respect to one another. The threshold distance between the turnscan be a function of a wavelength of a low-loss eigenmode solution ofthe Earth-ionosphere waveguide. For instance, the turns of coil can bespaced at intervals much less than the wavelength of an eigenmode of theEarth-ionosphere waveguide, perhaps at intervals equal to or less than aquarter wavelength of the eigenmode solution. In an example, theelements can be arranged on distance scales between 0.01 k and 0.0001 k,where k is the eigenmode wavelength of a particular eigenmode solution.

To overcome the reactive forces that result from the close proximity ofthe turns, the helical coupler 1500 can be operated as a phased array inwhich the relative phases of the electromagnetic excitations generatedby each turn can be controlled. In an embodiment, the phase and/ormagnitude of the electromagnetic excitation generated by each turn canbe controlled by controlling the phase and/or magnitude of power thatdrives each turn. In an example, each driven element can be coupled toan amplifier that can control the phase and/or magnitude of the powerprovided to each driven element. Additionally and/or alternatively, thehelical coupler 1500 can include capacitor and/or inductor banks thatcan be used to control the phase and/or magnitude of the power that isprovided to each driven element.

To further increase the gain of the supergain helical coupler 1500, thesupergain helical coupler 1500 can include parasitic elements that canshape the electromagnetic excitation generated by the helical coupler1500. Parasitic elements are not driven by a power source (i.e., notelectrically coupled to another element or source) to generate anelectromagnetic excitation, but rather can direct or modify anelectromagnetic excitation that is generated by another element (e.g., adriven element). By including parasitic elements, the supergain helicalcoupler 1500 can operate like a Yagi-array that includes a drivenelement, a reflector, and several directors. Like in a Yagi-array, theparasitic elements can operate as directors or reflectors that can shapethe electromagnetic excitation generated by the helical coupler 1500. Itcould be demonstrated using theory, simulation, and/or measurement thatthe gain could be increased using the helical coupler 1500 where atleast one element of the coupler is acting as a reflector or director.In particular, to increase the gain of the supergain helical coupler1500, the parasitic elements can further shape the electromagneticexcitation to be more directed.

In an embodiment, a length of a parasitic element, an orientation of theparasitic element, and/or the parasitic element's distance from thedriven element can determine how the parasitic element shapes theelectromagnetic excitation. Accordingly, the properties of a parasiticelement can be determined such that the parasitic element can increasethe directivity of the helical coupler 1500. For instance, it can bedetermined that the electromagnetic excitation of the helical coupler1500 can include electromagnetic excitations that are e.g. 180° out ofphase with respect to one another. To generate these excitations, ratherthan driving two elements separately, one element can be a parasiticelement with parameters such that when placed near a driven element, theparasitic element can generate an electromagnetic excitation that is180° out of phase with respect to the electromagnetic excitationgenerated by the driven element. That is, the parasitic element isdriven at anti-resonance sympathetically from the driven element.

In an example, the parasitic elements can be turns of the helicalcoupler 1500 that are not directly driven by a power source. In anotherexample, the parasitic elements can be stand-alone elements that aredisposed near one or more turns of the helical coupler 1500. Otherexamples of parasitic elements are also possible.

In an embodiment, the efficiency of the supergain helical coupler 1500can be increased by designing the helical coupler as an electricallysmall array. Generally, a radiating array is electrically small if thearray's dimensions are small compared to the wavelength. Morespecifically, a radiating array can be considered electrically small ifk×a<1.0, where k=2π/λ, and “a” is a radius of a sphere thatcircumscribes the radiating element as its “maximum dimension” measuredfrom its center.

In an implementation of the electrically small helical coupler 1500, theturns of the helical coupler 1500 can be electrically small. In anotherimplementation, in addition to being electrically small, the turns ofthe helical coupler 1500 can be resonant elements. Using electricallysmall resonant elements increases the efficiency of the helical coupler1500 because the driven elements of the coupler can be driven withoutthe use of large tuning reactances that can cause losses. This is due tothe fact that when two electrically small resonant elements are placedin close proximity of one another, their input reactances are smaller inmagnitude than the input reactances of below-resonance electricallysmall electric dipole antennas, and the resulting circulating currentscause an increase in I²R losses.

Another benefit of using resonant elements is the fact that they can beused as an effective passive director or reflector. In particular,shorting the input terminals of a resonant electrically small elementallows the element to operate as a passive director or reflector. Assuch, in an embodiment, the electrically small helical coupler 1500 canalso include one or more resonant elements with shorted input terminals.In an example, one or more turns of the helical coupler 1500 can beelectrically small resonant elements with shorted input terminals. Thus,the helical coupler 1500 can be an electrically small supergain arraythat includes one or more electrically small resonant driven elementsand one or more electrically small parasitic elements

In an embodiment, in order to achieve resonance, the helical coupler1500 can include one or more active and/or passive elements that canadjust the electrical length of the helical coupler 1500. For instance,in order to achieve resonance, the electrical length of the waveguidecoupler can be a fraction of the wavelength of the eigenmode to whichthe helical coupler 1500 is coupling. In an example, the electricallength of the antenna may be half of the wavelength of the eigenmode.However, given that the desired eigenmodes for wireless power couplingare low modes with lower frequencies (e.g., on scale of 10s of KHz), itmay not be physically feasible to build a helical coupler where theelectrical length and the physical length are comparable. Accordingly,the helical coupler 1500 can use the one or more active and/or passiveelements to adjust the electrical length of the helical coupler 1500such that the electrical length is much larger than the physicalstructure of the helical coupler 1500.

In an embodiment, the helical coupler 1500 can be a slow-wave structurein order to physically shorten the length of the structure whilemaintaining the electrical length of the element. In particular, in aslow-wave structure, the phase velocity of the a wave that ispropagating along the helical coupler 1500 is less than the speed oflight in a vacuum, which results in the structure having a physical sizethat is different than the structure's electrical size. In anembodiment, the slow-wave structure can be designed such that theelectrical size is larger than the physical size.

In an embodiment, the helical coupler 1500 can account for effects ofthe ionosphere on an electromagnetic excitation. For example, theeffects of the ionosphere that can affect an electromagnetic excitationinclude the Faraday effect in which the ionosphere can cause a planarwave to rotate, which results in the wave having a circularpolarization. Generally, a plane wave doesn't couple most of the powerto a mode that is circularly polarized, which decreases the efficiencyof the radiation. However, the electromagnetic excitation that isgenerated by the helical coupler 1500 can be elliptically polarized orapproximately elliptically polarized (e.g., circularly polarized). Thus,the excitation does not lose energy to the Faraday effect wheninteracting with the ionosphere.

Additionally, the helical coupler 1500 can account for the dynamic,inhomogeneous, and anisotropic nature of the Earth-ionosphere waveguide.In an example, the helical coupler 1500 can account for the dynamic andanisotropic nature of the ionosphere. As explained above, the status ofthe ionosphere (e.g., level of ionization) can depend primarily on solaractivity. Thus, the status of ionosphere can be location and timedependent. The helical coupler 1500 can also account for other elementsin the waveguide that can be affected by or affect the electromagneticgeneration generated by the helical coupler 1500, such as the climate,existing large electronic structures, and/or structures andtransmissions. Other examples are also possible.

In an embodiment, to account for the dynamic, inhomogeneous, andanisotropic nature of the waveguide, the helical coupler 1500 can beimplemented as a variable helical coil. Within examples, the variablehelical coupler 1500 can dynamically adjust one or more of itsparameters. The parameters that the variable coupler can adjust include(i) a height of each turn of coil, (ii) a wire diameter of each turn ofcoil, (iii) a winding diameter, (iv) a height of the helical coupler1500, (v) a number of turns N in the helical coupler 1500, (vi)conductor material, and (vii) the cross-section of the coils. Otherfeatures that the variable helical coupler 1500 can adjust include theshielding and orientation of the connections between the elements of thehelical coupler 1500.

In addition to adjusting the physical parameters of the helical coupler1500 to account for the current nature of the Earth-ionospherewaveguide, the helical coupler 1500 can also use discrete elements inorder to change the electric magnitude and/or phase relationships of thepower that is provided to each turn of coil of the helical coupler 1500,which, in turn, can adjust the electromagnetic excitation to account forthe status of the Earth-ionosphere waveguide.

In an embodiment, the variable helical coupler 1500 can determine one ormore properties of the waveguide in order to determine how to adjust toaccount for the dynamic, inhomogeneous, and anisotropic nature of thewaveguide. For instance, as explained above, the helical coupler 1500can continually or periodically determine the status of the ionospherein order to account for any changes in the ionosphere when generatingthe electromagnetic excitation. Based on the status of the ionosphere,the helical coupler 1500 can adjust one or more parameters of thehelical coupler 1500 in order to account for the status of theionosphere. Additionally and/or alternatively, the helical coupler 1500can determine to adjust the electric magnitude and/or phaserelationships of the power that is provided to each turn of coil. Theresulting helical coupler 1500 will have strategically-differing coildiameters and/or strategically-differing inter-loop spacing.

FIG. 17 illustrates a variable helical coupler 1700, according to anexample embodiment. As illustrated in FIG. 17, the helical coupler 1700can have strategically-differing parameters including (i) differingpitches, e.g., pitches 1702A and 1702B, (ii) differing wire diameters,e.g., wire diameters 1704A and 1704B, and (iii) differing windingdiameters, e.g., wire diameters 1706A and 1706B. As also illustrated inFIG. 17, the helical coupler 1700 can also have a height 1708.

In an embodiment, an excitation source may be used to provide thehelical coupler 1500 with energy with which the coupler can generate theelectromagnetic excitation. The excitation source can be electricallycoupled to the helical coupler 1500 either directly or indirectly. In anexample, the excitation source can be directly coupled to the helicalcoupler 1500 via one or more feed lines that directly drive one or moreturns of coil. In another example, the excitation source can beelectrically coupled to the helical coupler 1500 via magnetic and/orcapacitive coupling. For instance, to couple to the helical coupler 1500magnetically, the excitation source can include a resonant inductor thatcan inductively couple to the helical coupler 1500.

FIGS. 18A, 18B, 18C, and 18D illustrate different configurations of ahelical coupler coupled to one or more excitation sources, according toexample embodiments. FIG. 18A illustrates a first configuration 1800 ofa helical coupler. As illustrated in FIG. 18A, a helical coupler 1804 iscoupled to an excitation source 1802. In this configuration, theexcitation source 1802 includes a power source 1806 and an inductor1808. In an example, the inductor 1808 can be a coil of wire. As alsoillustrated in FIG. 18A, the helical coupler 1804 can include N turns ofcoil, such as turns E10A, E10B, and E10C. In this configuration, theturns of coil 1810A-1810C have identical dimensions.

FIG. 18B illustrates a second configuration 1820 of a helical coupler.As illustrated in FIG. 18B, a helical coupler 1824 is coupled to anexcitation source 1822. Like the configuration in FIG. 18A, theexcitation source 1822 includes a power source 1826 and an inductor1828, and the helical coupler 1824 includes N turns of coil. As alsoillustrated in FIG. 18B, the helical coupler 1824 includes a wiresegment 1832 between turns 1830B and 1830C. The wire segment is astraight wire connection that connects between two turns of coil, andcan be used to adjust the electrical phase of the power provided tocoils, adjust the phase velocity, and/or be used for phase matching. Inan example, the wire segment can have a length on the range of sub-feet,feet, tens of feet, hundreds of feet, etc.

FIG. 18C illustrates a third configuration 1840 of a helical coupler. Asillustrated in FIG. 18C, a helical coupler 1846 is coupled to excitationsources 1842 and 1844, where each of the excitation sources includes apower source (1848 and 1850 respectively) and an inductor (1852 and 1854respectively). As also illustrated in FIG. 18C, the helical couplerincludes N turns, e.g., turns 1856A, 1856B, and 1856C.

FIG. 18D illustrates a fourth configuration 1860 of a helical coupler.As illustrated in FIG. 18D, a helical coupler 1866 is coupled toexcitation sources 1862 and 1864, where each of the excitation sourcesincludes a power source (1868 and 1868 respectively) and an inductor(1872 and 1874 respectively). Within examples, the helical coupler 1866is a variable helical coupler that has variable parameters. Asillustrated in FIG. 18D, the variable helical coupler 1866 has includesN turns, e.g., turns 1876A, 1876B, 1876C, E76D, and 1876E, where theturns have strategically-differing pitch, wire diameters, and windingdiameters. For instance, each of the turns 1876A-1876E has a differentwire diameter.

The example configurations of the excitation source and the helicalcoupler provided in FIGS. 18A, 18B, 18C, and 18D, and the accompanyingdescription herein is for illustrative purposes only and should not beconsidered limiting. In an implementation, more than two excitationsources can be coupled to the helical coupler 1500.

FIG. 19 illustrates an isometric view of a helical coupler 1900,according to an example embodiment. As illustrated in FIG. 19, thehelical coupler 1900 is a variable helical coupler 1900 that includesstrategically-varying parameters. For instance, as illustrated in FIG.19, the turns of the helical coupler 1900 has varying diameters.

In an embodiment, the helical coupler 1900 can include supportinginfrastructure that allows the helical coupler 1900 to operate as atransmitter or receiver. For instance, the supporting infrastructure caninclude a power source that can provide electrical energy to atransmitter helical coupler 1900. An example power source can include arenewable energy source such as a solar or wind farm that can providethe helical coupler 1900 with electric energy. In another example, thesupporting infrastructure can include grid connections that transportenergy between an electric grid and helical coupler 1900. The gridconnections can include power or substation circuits that can align thephase and/or control the magnitude of the power that is provided to thegrid.

Additionally and/or alternatively, the supporting infrastructure caninclude an earth ground connection. The earth ground connection cancomprise an interconnected network of wires arranged on the surface ofthe Earth. The network of wires can extend radially from the base of thehelical coupler 1900, perhaps for a distance nearly equal to or equal tothe height of the helical coupler 1900. For example, the earth groundconnection can include conductors that are arranged in a spider-likelayout or that are arranged in fractals such that the conductors cover alarge amount of surface area with Earth. Increasing the amount ofsurface area that the conductors share with the Earth can decrease theimpedance of the earth ground connection, which decreases losses byimproving the quality of the earth ground connection. Other methods ofdecreasing the impedance of the earth ground connection can includeincreasing the depth to which the earth ground connections are driven,increasing the number of earth ground connections, increasing themoisture content of the soil, and improving the conductive mineralcontent of the soil. Additionally and/or alternatively, a counterpoisethat is supported above ground can be used as a ground plane.

FIG. 20 is a simplified block diagram illustrating a transmitterwaveguide coupler 2004 coupled to a power source 2002, according to anexample implementation. The transmitter waveguide coupler 2004 can beconfigured to generate an electromagnetic excitation that can couple toan eigenmode of a waveguide. As illustrated in FIG. 20, the waveguidecoupler 2004 can include an array of elements 2006 that can includedriven elements 2012 and parasitic elements 2014. The array of elements2006 can include resonators and/or coils that operate as an array ofdeep sub-wavelength spaced arrays of electric and magnetic dipolesources, connected with specific amplitude and phase relationshipsbetween the array elements.

As also illustrated in FIG. 20, the transmitter waveguide coupler 2004can include circuit elements 2008 that include, but are not limited to,capacitor banks 2016, inductor banks 2018, and amplifiers 2020. Withinexamples, the circuit elements 2008 may be arranged in the transmitterwaveguide coupler 2004 to achieve specific functionality such asimpedance matching, adjusting the electrical length of the waveguidecoupler, adjusting the magnitude and/or phase of the power provided toeach element, among other examples. Furthermore, the circuit elements2008 can be arranged to compensate for any inefficiencies (e.g.,radiation losses) in the transmitter waveguide coupler 2004. Forexample, if the load impedance and output impedance of an amplifier aredifferent, the utilization of the available energy may not be veryefficient. To compensate for the difference in impedance, one or morecircuit elements 2008 can be arranged to form an impedance matchingcircuit. Other possible circuit elements include but are not limited toinclude switches (e.g., solid state switches, vacuum tubes, mercuryswitches, etc.), transmission lines, open or short circuited stubs,transformers, strongly or weakly coupled magnetic or electricresonators.

Within examples, the power source 2002 may include a source of powersuch as a generator. As explained above, the power source 2002 can bedirectly or indirectly coupled to the waveguide coupler. Additionally,the power source 2002 can include one or more resonant elements thatallow the power source 2002 to wirelessly, via its near field, coupleenergy inductively and/or capacitively to the transmitter waveguidecoupler 2004.

In an embodiment, in order to generate an electromagnetic excitation,the transmitter waveguide coupler 2004 can substantially synthesizecurrents on lossy conducting mediums (e.g., the ionosphere and theterrestrial layer). By doing so, the transmitter waveguide coupler 2004can generate an electromagnetic excitation that is not a radiated wave,but rather is a guided wave in the Earth-ionosphere waveguide.

The waveguide coupler 2004 can excite a single eigenmode of theEarth-ionosphere waveguide or can excite a specific set of eigenmodeswithout exciting others. Additionally, the electromagnetic excitation ofthe waveguide coupler 2004 can overlap or substantially overlap aneigenmode of the Earth-ionosphere waveguide. An electromagneticexcitation that substantially overlaps an eigenmode has a thresholdoverlap integral with the eigenmode. The threshold overlap integral canbe expressed as a percentage or ratio. For example, the thresholdoverlap integral may be 50%. Other examples are possible.

FIG. 21 is a flowchart illustrating an example method. As with the otherexample methods described above and in accordance with exampleembodiments, the method illustrated in FIG. 21 may be implemented usingone or more transmitter apparatuses or devices, one or more receiverapparatuses or devices, and one or more computing systems configured forexecuting instructions for carrying out various steps and functionsdescribed herein.

At step 2102, one or more eigenmodes of a spherical waveguide cavityencompassed by two boundaries are computed. This can be one according toone or another of the techniques described above.

At step 2104, at least one of the one or more eigenmodes is chosen to beused in wireless energy transmission in the spherical waveguide cavity.

Finally, at step 2106, energy is coupled wirelessly into the sphericalwaveguide cavity, wherein the coupling comprises causing an electricallysmall array of coupling elements to generate an electromagneticexcitation that overlaps the at least one eigenmode.

In accordance with example embodiments, computing the one or moreeigenmodes may be based on a mathematical model that incorporatesrespective properties of the two boundaries.

In accordance with example embodiments, a first one of the boundaries isa terrestrial surface and a second one of the boundaries is anionospheric layer. For this arrangement, the respective properties ofthe terrestrial surface are electrical properties, and the respectiveproperties of the ionospheric layer are properties based on plasmaphysics of the ionospheric layer.

III. Example Systems and Methods for Global Transmission of Power inEigenmodes of the Earth-Ionosphere Waveguide

A single antenna array having supergain properties and/or incorporatingappropriate supergain antenna design aspects may serve as a singlewireless power transmission (excitation) station or “launcher” forcoupling into one or more low-loss eigenmodes of the Earth-ionospherewaveguide. The helical coupler described above can act as such asupergain array. By deploying multiple helical couplers and coordinatingtheir operation, global or nearly global wireless power distribution maybe achieved. In such scenarios, the entire volume of theEarth-ionosphere waveguide may support standing waves of one or moreeigenmodes such that the power density at any point may be tapped at alevel commensurate with a receiving antenna's electrical size andcoupling efficiency.

In accordance with example embodiments, the helical coupler can be usedfor wireless power transmission in the Earth-ionosphere waveguide. Inthis embodiment, the helical coupler can operate as a transmitter togenerate an electromagnetic excitation that can be guided along one ormore low-loss eigenmodes of the Earth-ionosphere waveguide.Additionally, the helical coupler can operate as a receiver to receivethe guided electromagnetic excitation.

FIG. 22 is a simplified block diagram illustrating a wireless powersystem 2200 that can be used for wireless power transmission in theEarth-ionosphere waveguide, according to an example implementation. Asillustrated in FIG. 22, the wireless power system 2200 can include apower source 2202, a transmitter helical coupler 2204, a receiverhelical coupler 2206, and a controller 2216.

In an embodiment, the transmitter 2204 and the receiver 2206 can bestrategically disposed within the waveguide cavity in order towirelessly transmit power between two locations. Determining a locationin which to install a transmitter or receiver can be based on propertiesof the waveguide between the two locations. Such properties includeproperties of the ionosphere and the terrestrial surface, e.g., aconductivity and permittivity of the ionosphere and the terrestrialsurface between the two locations. Other factors can also includelocations of natural or manmade obstacles that can interfere with theoperation of a helical coupler. Additional constraints on the locationscan include factors such as available area to install the transmittersand receivers, etc.

In an embodiment, the transmitter helical coupler 2204 can receiveinstructions to transmit a certain amount of energy to the receiverhelical coupler 2206. For instance, the controller 2216 can receiveinstructions or determine to send the certain amount of energy from thetransmitter helical coupler 2204 to the receiver helical coupler 2206.

In an embodiment, to transmit the energy to the receiver helical coupler2206, the transmitter helical coupler 2204 can generate a traveling wavethat is guided along one or more low-loss eigenmodes of theEarth-ionosphere waveguide. In order to generate an electromagneticexcitation that overlaps the one or more low-loss eigenmodes, thecontroller 2216 can determine the one or more low-loss eigenmodes thatwill be used for the transmission. The low-loss eigenmodes can be knownto the controller 2216, such as previously determine low-losseigenmodes, or can be determined using the methods described herein.

Once the one or more eigenmodes are determined, the controller 2216 candetermine a status of the Earth-ionosphere waveguide along the one ormore eigenmodes. Determining the status of the Earth-ionospherewaveguide can include determining a status or condition of theionosphere between the transmitter helical coupler 2204 and the receiverhelical coupler 2206 at the time of generating the electromagneticexcitation. For instance, a status of the ionosphere can include a levelof ionization and a height of ionosphere (e.g., the D layer).Additionally and/or alternatively, determining the status of theEarth-ionosphere waveguide can include determining a status or conditionof the terrestrial surface, such as the conductivity and/or permittivityof the surface.

Based on the determined status of the Earth-ionosphere waveguide, thecontroller 2216 can determine one or more parameters of the transmitterhelical coupler 2204. The one or more parameters of the transmitterhelical coupler 2204 can include a magnitude and phase of a powerprovided to each element of the transmitter helical coupler 2204.Additionally and/or alternatively, if the transmitter helical coupler2204 is a variable helical coupler, the one or more parameters caninclude parameters of the variable helical coupler, such as a pitch,wire diameter, wire cross-section, etc. In particular, the controller2216 can determine the one or more parameters such that the generatedelectromagnetic excitation overlaps or substantially overlaps one ormore low-loss modes of the Earth-ionosphere waveguide.

Once the parameters are determined, the controller 2216 can operate thetransmitter helical coupler 2204 at the determined parameters to send apulse or low energy signal to the receiver helical coupler 2206 todetermine whether the overlap integral between the pulse and theeigenmodes is greater than or equal to a predetermined efficiencythreshold. If the overlap integral is greater than the threshold, thenthe controller 2216 will proceed to operate the helical coupler 2204 atthe determined parameters to send the energy to the receiver helicalcoupler 2206. Otherwise, the controller 2216 can determine a differentset of one or more eigenmodes, and can repeat the steps described aboveuntil the overlap integral of a pulse is greater than or equal to thepredetermined efficiency threshold.

By performing these steps, the controller 2216 can effectively accountfor the status of the Earth-ionosphere waveguide when determining how togenerate and transmit the electromagnetic excitation that includes thecertain amount of energy. Then, to receive the transmitted energy, thereceiver helical coupler 2206 can receive the traveling wave (i.e., theelectromagnetic excitation). The receiver helical coupler 2206 can thensupply the received energy to a load (e.g., a grid). Within examples,the wireless power system 2200 can transmit power on the scale ofkilowatts, megawatts, gigawatts, or terawatts.

In order for the wireless power system 2200 to be used for wirelesspower transmission, the transmitter and receiver helical couplers 2204,2206 must be able to exchange large amounts of energy. In part due topower limitations of a single transmitter or receiver helical coupler, asingle helical coupler operating as a transmitter or receiver may not beable to handle enough energy to be considered useful for powertransmission.

To overcome the power limitations of a single helical coupler, thetransmitter helical coupler 2204 and the receiver helical coupler 2206can each include an array of helical couplers. By including an array ofhelical couplers, the transmitter helical coupler 2204 and the receiverhelical coupler 2206 can handle an amount of energy suitable for powertransmission. The array of helical couplers is described below in thecontext of the transmitter helical coupler 2204. However, thedescription can also be applicable to the receiver helical coupler 2206.

FIGS. 23A and 23B illustrate different configurations of a transmitterhelical coupler that includes an array of helical couplers, according toan example embodiment. In the configuration illustrated in FIG. 23A, thetransmitter coupler 2304 is an array of a plurality of helical couplersthat are coupled to a power source 2302. For illustration purposes, thetransmitter helical coupler 2304 is an array of two helical couplers2304 and 2306. However, the transmitter helical coupler 2304 can includeX helical couplers, where X is on the scale of tens or hundreds. Thehelical couplers of an array, e.g., array 2300, can be identical ordifferent helical couplers.

In the configuration illustrated in FIG. 23B, a transmitter helicalcoupler 2310 can include an array of helical couplers each coupled to arespective power source. As illustrated in FIG. 23B, each element of thearray includes a power source, e.g., power sources 2316 and 2320, and ahelical coupler, e.g., helical couplers 2318 and 2322. Other exampleconfigurations of transmitter helical couplers are also possible.

In an embodiment, the description of a single transmitter helicalcoupler transmitting energy to a single receiver helical coupler can beexpanded to a network of geographically distributed transmitter helicalcouplers and receiver helical couplers. In this embodiment, the networkof transmitter helical couplers and receiver helical couplers cantransmit power on a large scale, perhaps on a global scale.

To achieve power transmission on a global scale, the transmitter helicalcouplers and the receiver helical couplers can be strategically placedwithin the Earth-ionosphere waveguide. Similar to determining thelocation of a single helical coupler, determining a location for each ofthe helical couplers in the network can be based on one or moreproperties of the Earth-ionosphere waveguide and on properties of one ormore low-loss eigenmodes that could be used for power transmission.

As explained above, to be able to handle an amount of energy suitablefor power transmission, a transmitter can include an array of helicalcouplers. As such, the transmitter in a global wireless power system canbe operated as a phased array in which a magnitude and phase of anelectromagnetic excitation generated by each of the helical couplers canbe controlled to produce a desired electromagnetic excitation oftransmitter. And the electromagnetic excitation generated by a helicalcoupler can be controlled by adjusting one or more parameters of thehelical coupler. Accordingly, the electromagnetic excitation generatedby each array element (i.e., helical coupler) can be controlled in orderto generate an overall electromagnetic excitation that overlaps alow-loss eigenmode of the Earth-ionosphere waveguide. Effectively, acontroller of the system can control each element of each helicalcoupler in the array of helical couplers in order to cause thetransmitter to generate a particular electromagnetic excitation.

In an embodiment, the global wireless system can transmit power from oneor more transmitters to one or more receivers via one or more low-losseigenmodes of the Earth-ionosphere waveguide. To do so, the transmitterscan generate traveling waves in the Earth-ionosphere waveguide, asdescribed above. Each traveling wave can carry a certain amount ofenergy and can be directed to one or more receivers.

Additionally and/or alternatively, the geographically distributed arrayscan be operated with controlled inter-array magnitudes and phases so asto set up standing waves in the Earth-ionosphere. In such scenarios, theentire volume of the whole Earth-ionosphere waveguide can support thestanding waves overlapping one or more low-loss eigenmodes such that thepower density at any point can be tapped by a receiver. Morespecifically, the transmitters can be operated to electromagneticexcitations that interact to form one or more standing waves thatoverlap one or more low-loss eigenmodes of the Earth-ionospherewaveguide. Within examples, the locations of where the transmitters areinstalled in the Earth-ionosphere waveguide can depend on where thenodes or anti-nodes of a standing wave of a low-loss eigenmode arelocated.

Once the standing waves are formed, each standing wave oscillatesbetween a node and an anti-node. The transmitters can increase ordecrease the amount of power in a standing wave by continually orperiodically adjusting the amount of energy coupled into theEarth-ionosphere waveguide. Furthermore, once the standing waves areformed, each standing wave can be tapped by a receiver in order toextract energy from the electromagnetic excitation at a levelcommensurate with a receiver's electrical size and couple efficiency.The receivers can be disposed at particular locations with respect tothe nodes or anti-nodes of the established standing waves at locations.For example, some receivers can be disposed at locations where thestanding the standing waves peak.

FIG. 24 illustrates example transmitters 2400A, 2400B, and 2400Cgenerating standing waves in the Earth-ionosphere waveguide 2402,according to an example embodiment. As illustrated, each of thetransmitters feeds the standing waves, conceptually represented as wavepatterns (e.g., pattern 2404), at different locations. In an embodiment,the transmitters can generate the standing waves such that the waves canfollow a specific path in the Earth-ionosphere waveguide 2402. One ormore receivers that are also located in the Earth-ionosphere waveguide2402 can tap into the standing waves to extract energy.

FIG. 25 is a flowchart illustrating an example method. Again, as withthe other example methods described above and in accordance with exampleembodiments, the method illustrated in FIG. 25 may be implemented usingone or more transmitter apparatuses or devices, one or more receiverapparatuses or devices, and one or more computing systems configured forexecuting instructions for carrying out various steps and functionsdescribed herein.

At step 2502, a computation is made of one or more eigenmodes of aspherical waveguide having a waveguide cavity bounded by the terrestrialsurface of the Earth and the ionosphere of the Earth, the computationbeing based on a mathematical model of the spherical waveguide thatincorporates electrical properties of the terrestrial surface of theEarth and the ionosphere of the Earth.

And at step 2504, coupling energy into the one or more eigenmodes of thespherical waveguide cavity, including causing the phased array ofelectrically small waveguide couplers to generate an electromagneticexcitation that overlaps the one or more eigenmodes.

In accordance with example embodiments, the electromagnetic excitationmay be a composite electromagnetic excitation, and coupling energy intothe spherical waveguide cavity (in step 2504) may entail determining atleast one of a desired amplitude, frequency, and phase of respectiveelectromagnetic excitations generated by two or more of the electricallysmall waveguide couplers, and determining respective operationalconfigurations for the two or more electrically small waveguide couplersbased on the at least one of the desired amplitude, frequency, andphase. Then, the two or more electrically small waveguide couplers maybe operated at the respective operational configurations to generate therespective electromagnetic excitations.

In further accordance with example embodiments, a phase relation betweenthe respective electromagnetic excitations of any two of the two or morewaveguide couplers may be based on relative geographic locations of thetwo waveguide couplers.

In accordance with example embodiments, each of the electrically smallwaveguide couplers may include a respective array of helical waveguidecouplers. In further accordance with example embodiments, theelectromagnetic excitation may be a composite electromagneticexcitation. With this arrangement, coupling energy into the sphericalwaveguide cavity (in step 2504) may entail causing two or more of thehelical waveguide couplers of two or more of the respective arrays ofthe helical waveguide couplers to generate respective electromagneticexcitations. In particular, the respective electromagnetic excitationsmay combine to form the composite electromagnetic excitation.

In further accordance with example embodiments, causing two or more ofthe helical waveguide couplers of two or more of the respective arraysof the helical waveguide couplers to generate respective electromagneticexcitations may entail determining at least one of a desired amplitude,frequency, and phase of the respective electromagnetic excitations anddetermining respective operational configurations for the two or more ofthe helical waveguide couplers based on the at least one of the desiredamplitude, frequency, and phase. Then, the two or more of the helicalwaveguide couplers may be operated at the respective operationalconfigurations to generate the respective electromagnetic excitations.

In accordance with example embodiments, the method in FIG. 25 mayfurther entail receiving information indicative of a reflected signalfrom a load that is extracting energy from the electromagneticexcitation, and based on the information indicative of the signal,detecting a presence of the load in the spherical waveguide.

IV. Example Systems and Methods for Detection and Determining Locationof Loads in the Earth-Ionosphere Waveguide

In accordance with example embodiments, the relative phases of arrays ofantennas may be dynamically adjusted in order to detect locations atwhich power is being extracted, and to measure how much is beingextracted. That is, detecting the presence and locations of receiverloads.

In an embodiment, a particular transmitter can generate a traveling wavethat could be transmitted to a particular receiver. Alternatively, thetransmitter can feed into or establish a standing wave in the waveguide.In some examples, the transmitter can periodically switch betweencontributing a standing wave and sending a traveling wave to aparticular receiver.

In an embodiment, controlling each of the transmitters to generate anelectromagnetic excitation travels in a particular direction allows thewireless power system to steer power in the Earth-ionosphere waveguide.Specifically, the controller of the system can cause the transmitters togenerate electromagnetic excitations that overlap to form an overallelectromagnetic excitation that travels in a particular direction. Thatis, the controller of the system can generate electromagneticexcitations (e.g., standing and traveling) waves with nodes andantinodes (also called nulls) at specific locations. As such, thecontroller has the ability to steer the overall electromagneticexcitation, and thus the power, towards or away from any location.

In an embodiment, the ability to steer power allows the wireless powersystem to detect the location of a load that is coupled to the system(whether a stationary or moving load). Within examples, various methodsto detect the presence of a load could be based on transmission linetheory. Transmission line theory is applicable here because waveguidesare a special form of transmission lines, and could be modeled as such.In this analogy, the low-loss eigenmodes along which an electromagneticexcitation is guided is analogous to a transmission line.

In an embodiment, the system could detect the presence of a load bydetecting a reflected signal from the load. For instance, the systemcould transmit a signal (e.g., a pulse) along a low-loss eigenmode. Ifthere is a load coupled to the eigenmode, the load will receive thepulse. Due to impedance mismatches, a reflected signal will betransmitted from the load to the transmitter, and the system can detectthe reflected signal. By detecting the reflected signal, the system candetect the presence of the load that is coupled to the eigenmode.

Additionally, the system could determine a location of the detectedload. In an embodiment, the system can determine the location based onthe reflected signal from the load. Using the transmission line analogy,determining a location of the load is similar to finding a location of aload in a network of transmission lines. For example, after detecting aload, the system can steer power towards the load until the steeredpower reaches a steady state and a higher “standing wave ratio” (SWR)will be present. Once the power reaches a steady state, the SWR data canbe plotted as a function of distance based on the reflection returntime. Analyzing the SWR data can indicate approximately the location ofthe load. Additionally and/or alternatively, the magnitudes and phasesand locations of the resulting standing waves can be analyzed todetermine the location of the load.

The accuracy of this method of detection increases the more sources anddifferent starting locations from which the pulse signals are sent.Essentially, this method is a linear algebra problem where the unknownis the location of the receiver, and the known values are thecharacteristics of the pulse and return signals. With enough signals, abest fit of the data can be taken and an accurate measurement can bemade.

FIG. 26 is a flowchart illustrating an example method. Again, as withthe other example methods described above and in accordance with exampleembodiments, the method illustrated in FIG. 26 could be implemented outusing one or more transmitter apparatuses or devices, one or morereceiver apparatuses or devices, and one or more computing systemsconfigured for executing instructions for carrying out various steps andfunctions described herein.

FIG. 26 is a flowchart illustrating an example method, according to anexample embodiment. As step 2602, the method can involve computing oneor more eigenmodes of a spherical waveguide having a waveguide cavitybounded by the terrestrial surface of the Earth and the ionosphere ofthe Earth, the computation being based on a mathematical model of thespherical waveguide that incorporates electrical properties of theterrestrial surface of the Earth and the ionosphere of the Earth.

At step 2604, the method can involve coupling energy into the one ormore eigenmodes of the spherical waveguide cavity by generating anexcitation via two or more waveguide couplers of a plurality ofwaveguide couplers of a phased waveguide coupler array, wherein eachwaveguide coupler comprises an electrically small array ofwaveguide-coupling elements, and wherein each waveguide coupler islocated at a respective geographic location;

At step 2606, the method can involve detecting a presence of a powerload in a geographic region in the spherical waveguide. In particular,the wireless power system can detect the presence of a power load thatis coupled to the waveguide coupler and that is extracting energy fromthe electromagnetic excitation generated by the wireless power system.In one example, detecting a presence of the power load in the waveguidecould involve receiving information indicative of a reflected signalfrom the load that is extracting energy from the electromagneticexcitation. Then, based on the information indicative of the signal, thewireless power system can detect the presence of the load. For instance,the reflected signal could be at least a threshold different from anexpected reflected signal. In another example, detecting a presence ofthe power load in the waveguide could involve receiving, from at leastone strategically placed field probe, information indicative of a fieldstrength of the electromagnetic excitation at one or more givenlocations. Then, based on the information, the system could detect apresence of a load in the spherical waveguide. In yet another example,the system could analyze the received signals at different receivers,perhaps to determine information indicative of at least one of amagnitude and phase of a respective received signal. Then based on theinformation, the system could detect a presence of a load in thespherical waveguide. For instance, there could be a threshold differencebetween the received signals and expected received signals.

At step 2606, the method can involve adjusting at least one of anamplitude, frequency, or phase of the generated excitation of the two ormore waveguide couplers so as to increase a power level at thegeographic region. That is, the wireless power system can steer powertowards a particular location. For instance, the wireless power systemcan steer power towards the particular location in order to providepower to an authorized receiver. More specifically, the authorizedreceiver could requesting additional power. Additionally and/oralternatively, the system could detect that the power is below athreshold level in a particular area.

At step 2608, the method then involves determining a location of thepower load. In an example, determining the location of the load couldinvolve determining that the power level has reached a steady state, andbased on at least one of a magnitude, phase, and location of standingwaves that result in the steady state power level, determining a preciselocation of the load. In another example, determining the location ofthe load could involve detecting one or more respective reflectedsignals at the two or more waveguide couplers of the plurality ofwaveguide couplers, and based on the one or more respective reflectedsignals, using triangulation to determine the precise location of theload. In yet another example, determining the location of the load couldinvolve determining a magnitude and phase of the electromagneticexcitation at one or more locations; and based at least on (i) themagnitude and phase of the electromagnetic excitation, and (ii) one ormore or reflected signals received at the two or more waveguide couplersof the plurality of waveguide couplers, precisely determining thelocation of the power load.

V. Example Systems and Methods for Controlling Power Levels atDesignated Locations in Eigenmodes of Earth-Ionosphere Waveguide

In accordance with example embodiments, the relative phases of arrays ofantennas may be dynamically adjusted in order to control a level ofpower that is available at any given geographic location. This can bedone by controlling the location of nulls in standing waves, forexample. Doing so may allow wireless power delivery to be denied at aparticular location (e.g., where illegitimate or unauthorized tapping isoccurring) and/or to be enhanced (e.g., where particular legitimateneeds are deemed underserved).

In some examples, before determining a location of a load, the systemcan determine whether the load is an authorized user or a parasiticload. In such scenarios, the system can determine the location of theparasitic loads in order to steer power away from the parasitic loads.In one example, determining whether the load is parasitic can involvedetecting unexpected reflection signals from particular loads indicatingthat either an authorized load is using an unexpected amount of power orthat a parasitic load is coupled to the system.

Once the detection occurs, the system or an operator of the system cancommunicate with known (e.g., authorized) users to determine whetherthey have been using the unexpected amount of power. If the authorizeduser can account for the unexpected use, then it is likely that aparasitic load does not exist. Conversely, if the authorized user doesnot account for the unexpected use, then a parasitic load could existand further actions could be taken.

In an embodiment, if a parasitic load is detected, the system canrespond by steering power away from the location of the parasitic load.For example, the system can adjust the location of the nulls in thestanding and/or traveling waves from which the parasitic load isreceiving power. As explained above, adjusting the location of the nullscan involve adjusting the electromagnetic excitation generated by one ormore of the transmitters using methods described herein.

In an embodiment, the system can actively monitor for parasitic loadsusing an active feedback loop. The feedback loop can continuously orperiodically measure the reflected signals to detect any unexpected use,which can be indicative of unauthorized use. Once the system detects theunexpected use, the system can adjust the standing and/or travelingwaves until the feedback loop indicates that the reflected signals havereturned to expected levels. Within examples, the feedback loop can relyon additional electric and/or magnetic field measurements outside ofthose inherent in the transmitters themselves in order to detectunexpected use.

FIG. 27 is a flowchart illustrating an example method. As with the otherexample methods described above and in accordance with exampleembodiments, the method illustrated in FIG. 27 could be implemented outusing one or more transmitter apparatuses or devices, one or morereceiver apparatuses or devices, and one or more computing systemsconfigured for executing instructions for carrying out various steps andfunctions described herein. The computing system 100 illustrated in FIG.1 is an example of a computing system that could carry out steps andfunctions of the example methods. Instructions for execution by one ormore processors of the computing system could be stored as software,hardware, and/or firmware in a non-transitory computer-readable medium.Thus, steps, operations, and/or functions described herein may becarried out by the computing system when the instructions are executedby one or more processors of the one or more computing systems.

More particularly, the method illustrated in FIG. 27 may involve aphased waveguide coupler array that includes a plurality of waveguidecouplers, where each waveguide coupler includes an electrically smallarray of waveguide-coupling elements, such as electric dipole and/ormagnetic (helical) dipole antennas. Each of the plurality of waveguidecouplers may be located at different geographic locations. The phasedwaveguide coupler array—i.e., the plurality of waveguide couplers—mayact as described above to wirelessly transmit power in theEarth-ionosphere waveguide by coupling to one or more low-losseigenmodes.

As step 2701, one or more eigenmodes of the Earth-ionosphere waveguideare compute, where the computation is based on a mathematical model ofthe spherical waveguide that incorporates electrical properties of theterrestrial surface of the Earth and the ionosphere of the Earth.

At step 2704, energy is coupled into the one or more eigenmodes of thespherical waveguide cavity by generating an excitation via two or morewaveguide couplers of a plurality of waveguide couplers of a phasedwaveguide coupler array. As described above, each waveguide coupler mayinclude an electrically small array of waveguide-coupling elements.

Finally, at step 2706, at least one of an amplitude, frequency, or phaseof the generated excitation of the two or more waveguide couplersadjusted in such was as to control a power level at a given geographiclocation due to the energy coupled into the spherical waveguide. Thegiven geographic location is taken to be different from any of the twoor more waveguide couplers.

In accordance with example embodiments, the one or more eigenmodes mayform standing waves of an electric and magnetic vector field. In thiscase, adjusting the at least one of the amplitude, frequency, or phaseof the generated excitation of the two or more waveguide couplers mayentail determining a relative change in location of at least one powernull of the standing waves with respect to the given geographic locationsuch that the power level at the given geographic location will changeby a specified amount. Then the at least one of the amplitude,frequency, or phase of the generated excitation of the two or morewaveguide couplers may be adjusted to cause the relative change inlocation of the at least one power null of the standing waves.

In further accordance with example embodiments, adjusting the at leastone of the amplitude, frequency, or phase of the generated excitation ofthe two or more waveguide couplers may entail determining locations ofthe at least one power null of the standing waves as a function ofrelative phases between the two or more waveguide couplers, anddetermining power levels at the given geographic location as a functionof position of the at least one power null with respect to the givengeographic location.

In accordance with example embodiments, the one or more eigenmodes mayform standing waves of an electric and magnetic vector field, and themethod of FIG. 27 may further entail detecting an unauthorized powerload at the given geographic location. With this arrangement, adjustingthe at least one of the amplitude, frequency, or phase of the generatedexcitation of the two or more waveguide couplers may entail adjustingthe at least one of the amplitude, frequency, or phase of the generatedexcitation of the two or more waveguide couplers to steer at least onepower null of the standing waves from a first location to a secondlocation, where the second location is closer to the given geographiclocation than the first location.

In further accordance with example embodiments, detecting theunauthorized power load at the given geographic location may entaildetecting a power load at the given geographic location, and determiningthat there is no authorized power load at the given geographic location.For example the system detecting the power load and location may consulta database of authorized loads at various locations.

In accordance with example embodiments, the one or more eigenmodes formstanding waves of an electric and magnetic vector field, and the methodof FIG. 27 may further entail determining that the power level at thegiven geographic location due to the energy coupled into the sphericalwaveguide is below a threshold level. With this arrangement, adjustingthe at least one of the amplitude, frequency, or phase of the generatedexcitation of the two or more waveguide couplers may entail adjustingthe at least one of the amplitude, frequency, or phase of the generatedexcitation of the two or more waveguide couplers to steer at least onepower null of the standing waves from a first location to a secondlocation, where the second location is further away from the givengeographic location than the first location.

In further accordance with example embodiments, determining that thepower level at the given geographic location due to the energy coupledinto the spherical waveguide is below a threshold level may entaildetecting a power load at the given geographic location, and receivingan indication from the power load of the power level at the givengeographic location. For example, the power load may communicate withthe transmitting system by way of a communications network.

The present disclosure is not to be limited in terms of the particularembodiments described in this application, which are intended asillustrations of various aspects. Many modifications and variations canbe made without departing from its spirit and scope, as will be apparentto those skilled in the art. Functionally equivalent methods andapparatuses within the scope of the disclosure, in addition to thoseenumerated herein, will be apparent to those skilled in the art from theforegoing descriptions. Such modifications and variations are intendedto fall within the scope of the appended claims.

The above detailed description describes various features and functionsof the disclosed systems, devices, and methods with reference to theaccompanying figures. In the figures, similar symbols typically identifysimilar components, unless context dictates otherwise. The illustrativeembodiments described in the detailed description, figures, and claimsare not meant to be limiting. Other embodiments can be utilized, andother changes can be made, without departing from the spirit or scope ofthe subject matter presented herein. It will be readily understood thatthe aspects of the present disclosure, as generally described herein,and illustrated in the figures, can be arranged, substituted, combined,separated, and designed in a wide variety of different configurations,all of which are explicitly contemplated herein.

With respect to any or all of the ladder diagrams, scenarios, and flowcharts in the figures and as discussed herein, each block and/orcommunication may represent a processing of information and/or atransmission of information in accordance with example embodiments.Alternative embodiments are included within the scope of these exampleembodiments. In these alternative embodiments, for example, functionsdescribed as blocks, transmissions, communications, requests, responses,and/or messages may be executed out of order from that shown ordiscussed, including substantially concurrent or in reverse order,depending on the functionality involved. Further, more or fewer blocksand/or functions may be used with any of the ladder diagrams, scenarios,and flow charts discussed herein, and these ladder diagrams, scenarios,and flow charts may be combined with one another, in part or in whole.

A block that represents a processing of information may correspond tocircuitry that can be configured to perform the specific logicalfunctions of a herein-described method or technique. Alternatively oradditionally, a block that represents a processing of information maycorrespond to a module, a segment, or a portion of program code(including related data). The program code may include one or moreinstructions executable by a processor for implementing specific logicalfunctions or actions in the method or technique. The program code and/orrelated data may be stored on any type of computer readable medium suchas a storage device including a disk or hard drive or other storagemedium.

The computer readable medium may also include non-transitory computerreadable media such as non-transitory computer-readable media thatstores data for short periods of time like register memory, processorcache, and random access memory (RAM). The computer readable media mayalso include non-transitory computer readable media that stores programcode and/or data for longer periods of time, such as secondary orpersistent long term storage, like read only memory (ROM), optical ormagnetic disks, compact-disc read only memory (CD-ROM), for example. Thecomputer readable media may also be any other volatile or non-volatilestorage systems. A computer readable medium may be considered a computerreadable storage medium, for example, or a tangible storage device.

Moreover, a block that represents one or more information transmissionsmay correspond to information transmissions between software and/orhardware modules in the same physical device. However, other informationtransmissions may be between software modules and/or hardware modules indifferent physical devices.

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for providedfor explanatory purposes and are not intended to be limiting, with thetrue scope being indicated by the following claims.

What is claimed:
 1. A method comprising: transmitting, by a transmitterapparatus, electrical power into a spherical waveguide bounded by aterrestrial surface and an ionospheric layer, wherein the electricalpower is transmitted in an electromagnetic wave; computing one or moreeigenmodes of the spherical waveguide based on a mathematical model ofthe spherical waveguide that incorporates electrical properties of theterrestrial surface and plasma physics of the ionospheric layer;detecting the transmitted electromagnetic wave by a receiver apparatusremote from the transmitter apparatus; and determining a strength ofcoupling between the transmitted electromagnetic wave and the one ormore eigenmodes by measuring an amount of power received by the receiverapparatus in the detected electromagnetic wave.
 2. The method of claim1, wherein the terrestrial surface is the surface of the Earth and theionospheric layer is the ionospheric layer of the Earth.
 3. The methodof claim 1, wherein computing the one or more eigenmodes of thespherical waveguide comprises: numerically solving Maxwell's Equationsapplied to the mathematical model of the spherical waveguide;determining the one or more eigenmodes from the numerical solution inthe form of computed electric and magnetic field vectors at discretespatial points of the mathematical model of the spherical waveguide; anddetermining eigenfrequencies and propagation constants for the one ormore eigenmodes from the numerical solution.
 4. The method of claim 3,wherein numerically solving Maxwell's Equations applied to themathematical model of the spherical waveguide comprises computing anumerical simulation in at least one of: two spatial dimensions, orthree spatial dimensions.
 5. The method of claim 3, wherein transmittingthe electrical power into the spherical waveguide comprises transmittingthe electrical power at one or more of the eigenfrequencies.
 6. Themethod of claim 1, wherein transmitting the electrical power into thespherical waveguide comprises encoding particular information into thetransmitted electromagnetic wave, and where detecting the transmittedelectromagnetic wave by the receiver apparatus comprises detecting theencoded particular information.
 7. The method of claim 6, wherein theparticular information comprises a timing signature.
 8. The method ofclaim 1, wherein determining the strength of coupling between thetransmitted electromagnetic wave and the one or more eigenmodes bymeasuring the amount of power received by the receiver apparatus in thedetected electromagnetic wave comprises: measuring electric and magneticfield vectors of the detected electromagnetic wave; and comparing themeasured electric and magnetic field vectors with computed electric andmagnetic field vectors of the one or more eigenmodes at a spatiallocation in the mathematical model of the spherical waveguidecorresponding to that of the receiver apparatus.
 9. The method of claim1, further comprising: based on the determined strength of coupling,determining a statistical confidence that the detected electromagneticwave was coupled to the one or more eigenmodes during a time interval inwhich it was detected by the receiver apparatus.
 10. The method of claim1, wherein the one or more eigenmodes form standing waves of an electricand magnetic vector field, wherein the receiver apparatus comprises aplurality of receiver devices, each at a different remote location fromthe transmitter apparatus, and wherein determining the strength ofcoupling between the transmitted electromagnetic wave and the one ormore eigenmodes comprises: predicting relative strengths of couplingamong the plurality of receiver devices at each of the different remotelocations based on standing waves; and comparing relative powermeasurements among the plurality of receiver devices with the predictedrelative strengths of coupling.
 11. The method of claim 10, whereinpredicting relative strengths of coupling among the plurality ofreceiver devices at each of the different remote locations based onstanding waves comprises: computing a numerical simulation of thetransmitted electromagnetic wave based on a mathematical model oftransmission properties of the transmitter apparatus; computing adiscretized overlap integral between the one or more eigenmodes and thesimulated transmitted electromagnetic wave at simulated locationscorresponding to the different remote locations of the plurality ofreceivers; and comparing the discretized overlap integral with anempirical overlap integral derived from measurements of electric andmagnetic field vectors at the plurality of receiver devices.
 12. Themethod of claim 1, wherein the transmitter apparatus comprises aplurality of electrically connected waveguide coupling elements, whereintransmitting electrical power into the spherical waveguide by thetransmitter apparatus comprises transmitting electrical power by two ormore waveguide coupling elements of the plurality while maintainingrelative phases between the two or more waveguide coupling elements. 13.The method of claim 12, wherein the method further comprises: predictinga relative change in the coupling strength between the transmittedelectromagnetic wave and the one or more eigenmodes that would resultfrom one or more given changes in the relative phases between the two ormore waveguide coupling elements; predicting a change in the amount ofpower received by the receiver apparatus in the detected electromagneticwave based on the predicted relative change in coupling strength;adjusting the relative phases between the two or more waveguide couplingelements by the one or more given changes in the relative phases duringtransmission; and determining whether or not a measured change in powerreceived at the receiver apparatus is within a threshold of thepredicted change in received power.
 14. A system comprising: atransmitter apparatus; a receiver apparatus remote from the transmitterapparatus; and a computer apparatus having one or more processors andmemory storing instructions that, when executed by the one or moreprocessors, cause the system to carry out operations including: causingthe transmitter apparatus to transmit electrical power into a sphericalwaveguide bounded by the terrestrial surface of the Earth and theionospheric layer of the Earth, wherein the electrical power istransmitted in an electromagnetic wave; computing one or more eigenmodesof the spherical waveguide based on a mathematical model of thespherical waveguide that incorporates electrical properties of theterrestrial surface and plasma physics of the ionospheric layer; causingthe receiver apparatus to detect the transmitted electromagnetic wave;and determining a strength of coupling between the transmittedelectromagnetic wave and the one or more eigenmodes by measuring anamount of power received by the receiver apparatus in the detectedelectromagnetic wave.
 15. The system of claim 14, wherein computing theone or more eigenmodes of the spherical waveguide comprises: numericallysolving Maxwell's Equations applied to the mathematical model of thespherical waveguide; determining the one or more eigenmodes from thenumerical solution in the form of computed electric and magnetic fieldvectors at discrete spatial points of the mathematical model of thespherical waveguide; and determining eigenfrequencies and propagationconstants for the one or more eigenmodes from the numerical solution.16. The system of claim 15, wherein numerically solving Maxwell'sEquations applied to the mathematical model of the spherical waveguidecomprises computing a numerical simulation in at least one of: twospatial dimensions, or three spatial dimensions.
 17. The system of claim15, wherein causing the transmitter apparatus to transmit the electricalpower into the spherical waveguide comprises causing the transmitterapparatus to transmit the electrical power at one or more of theeigenfrequencies.
 18. The system of claim 14, wherein causing thetransmitter apparatus to transmit the electrical power into thespherical waveguide comprises causing the transmitter apparatus toencode particular information into the transmitted electromagnetic wave,and where causing the receiver apparatus to detect the transmittedelectromagnetic wave by the receiver apparatus comprises causing thereceiver apparatus to detect the encoded particular information.
 19. Thesystem of claim 18, wherein the particular information comprises atiming signature.
 20. The system of claim 14, wherein determining thestrength of coupling between the transmitted electromagnetic wave andthe one or more eigenmodes by measuring the amount of power received bythe receiver apparatus in the detected electromagnetic wave comprises:measuring electric and magnetic field vectors of the detectedelectromagnetic wave; and comparing the measured electric and magneticfield vectors with computed electric and magnetic field vectors of theone or more eigenmodes at a spatial location in the mathematical modelof the spherical waveguide corresponding to that of the receiverapparatus.
 21. The system of claim 14, wherein the operations furtherinclude: based on the determined strength of coupling, determining astatistical confidence that the detected electromagnetic wave wascoupled to the one or more eigenmodes during a time interval in which itwas detected by the receiver apparatus.
 22. The system of claim 14,wherein the one or more eigenmodes form standing waves of an electricand magnetic vector field, wherein the receiver apparatus comprises aplurality of receiver devices, each at a different remote location fromthe transmitter apparatus, and wherein determining the strength ofcoupling between the transmitted electromagnetic wave and the one ormore eigenmodes comprises: predicting relative strengths of couplingamong the plurality of receiver devices at each of the different remotelocations based on standing waves; and comparing relative powermeasurements among the plurality of receiver devices with the predictedrelative strengths of coupling.
 23. The system of claim 22, whereinpredicting relative strengths of coupling among the plurality ofreceiver devices at each of the different remote locations based onstanding waves comprises: computing a numerical simulation of thetransmitted electromagnetic wave based on a mathematical model oftransmission properties of the transmitter apparatus; computing adiscretized overlap integral between the one or more eigenmodes and thesimulated transmitted electromagnetic wave at simulated locationscorresponding to the different remote locations of the plurality ofreceivers; and comparing the discretized overlap integral with anempirical overlap integral derived from measurements of electric andmagnetic field vectors at the plurality of receiver devices.
 24. Asystem comprising: a transmitter apparatus; a receiver apparatus remotefrom the transmitter apparatus; and a computer apparatus having one ormore processors and memory storing instructions that, when executed bythe one or more processors, cause the system to carry out operationsincluding: causing the transmitter apparatus to transmit electricalpower into a spherical waveguide bounded by the terrestrial surface ofthe Earth and the ionospheric layer of the Earth, wherein the electricalpower is transmitted in an electromagnetic wave; computing one or moreeigenmodes of the spherical waveguide based on a mathematical model ofthe spherical waveguide that incorporates electrical properties of theterrestrial surface and plasma physics of the ionospheric layer; causingthe receiver apparatus to detect the transmitted electromagnetic wave;based on the determined one or more eigenmodes and an amount of powerreceived by the receiver apparatus in the detected electromagnetic wave,adjusting at least one of a frequency, amplitude, or phase of theelectrical power transmitted by the transmitter apparatus so as to causea predicted change in the amount of power received by the receiverapparatus in the detected electromagnetic wave; and determining whetheror not a measured change in power received at the receiver apparatus iswithin a threshold of the predicted change in received power.
 25. Thesystem of claim 24, wherein computing the one or more eigenmodes of thespherical waveguide comprises: numerically solving Maxwell's Equationsapplied to the mathematical model of the spherical waveguide;determining the one or more eigenmodes from the numerical solution inthe form of computed electric and magnetic field vectors at discretespatial points of the mathematical model of the spherical waveguide; anddetermining eigenfrequencies and propagation constants for the one ormore eigenmodes from the numerical solution.
 26. The system of claim 24,wherein causing the transmitter apparatus to transmit the electricalpower into the spherical waveguide comprises causing the transmitterapparatus to transmit the electrical power at one or more of theeigenfrequencies.
 27. The system of claim 24, wherein the transmitterapparatus comprises a plurality of electrically connected waveguidecoupling elements, wherein causing the transmitter apparatus to transmitelectrical power into the spherical waveguide comprises causing thetransmitter apparatus to transmit electrical power by two or morewaveguide coupling elements of the plurality while maintaining relativephases between the two or more waveguide coupling elements, and wherein,based on the computed one or more eigenmodes and the amount of powerreceived by the receiver apparatus in the detected electromagnetic wave,adjusting at least one of the frequency, amplitude, or phase of theelectrical power transmitted by the transmitter apparatus comprises:predicting a relative change in coupling strength between thetransmitted electromagnetic wave and the one or more eigenmodes thatwould result from one or more given changes in the relative phasesbetween the two or more waveguide coupling elements; predicting thechange in the amount of power received by the receiver apparatus in thedetected electromagnetic wave based on the predicted relative change incoupling strength; and adjusting the relative phases between the two ormore waveguide coupling elements by the one or more given changes in therelative phases during transmission.
 28. The system of claim 27, whereinthe one or more eigenmodes form standing waves of an electric andmagnetic vector field, and wherein predicting the change in the amountof power received by the receiver apparatus in the detectedelectromagnetic wave based on the predicted relative change in couplingstrength comprises: predicting relative changes in the electric andmagnetic field vectors of the standing waves at the receiver apparatus.29. The system of claim 27, wherein the receiver apparatus comprises aplurality of receiver devices, each at a different remote location fromthe transmitter apparatus, and wherein determining whether or not themeasured change in power received at the receiver apparatus is within athreshold of the predicted change in received power comprises: computinga first empirical overlap integral derived from measurements of electricand magnetic field vectors at the plurality of receiver devices prior toadjusting the relative phases between the two or more waveguide couplingelements; computing a second empirical overlap integral derived frommeasurements of electric and magnetic field vectors at the plurality ofreceiver devices after adjusting the relative phases between the two ormore waveguide coupling elements; and comparing a ratio of the first andsecond empirical overlap integrals with the predicted relative change incoupling strength between the transmitted electromagnetic wave and theone or more eigenmodes.